## 4 Numerical Simulation for CFRP-Strengthened Corroded RC Beams
### Numerical Simulation Model  
Simulations were carried out using the finite element software ABAQUS, and the separated model approach was chosen for finite element modeling. Considering that the corrosion of steel bars in the experiment was obtained via the accelerated corrosion method, it is assumed during simulation that steel bars are uniformly corroded. The loss of steel bar cross-sectional area caused by corrosion can be achieved through modifying the cross-sectional dimensions of the steel bar component. Concrete is simulated using C3D8R elements with a plastic damage model. Steel bars are simulated using T3D2 elements, while CFRP is simulated using C3D8R elements. The fatigue constitutive models for each material are calculated according to the formula described in Section 2. In addition, rigid pads are placed at the points of concentrated force loading and at the supports to avoid stress concentration. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Beam ID**} & \multirow{2}{*}{\(\mathbf{\eta}\) (\%)} & \multicolumn{2}{c}{**CFRP Strengthening Amount**} & \multirow{2}{*}{\
\begin{tabular}{c} **Fatigue Load** \\ **Pmin/Pmax (kN)** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Fatigue Life** \\ **N (10\({}^{4}\))** \\ \end{tabular} } \\ \cline{3-5}  & & & & & \\ \hline Beam No. 8 & 8.3 & 1 & 100/1.1 & 10/35 & 134.6 \\ Beam No. 10 & 15.0 & 1 & 100/1.1 & 10/35 & 61.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Test of strengthened beam. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Material**} & \multicolumn{2}{c}{**Compressive Strength**} & \multirow{2}{*}{**Tensile Strength**} & \multirow{2}{*}{**Elastic Modulus** \\  & & **(MPa)** & **(MPa)** & **(GPa)** \\ \hline Concrete & & 25.5 & & 29.8 \\ Steel bar & \(\phi\)12 & & 382 & 200 \\  & \(\phi\)6 & & 352.5 & 210 \\  & CFRP & & 4171 & 267 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mechanical properties of materials. Figure 2: Test beam size and reinforcement drawing (mm). Corrosion of steel bars can lead to degradation of the bond between steel bars and concrete. Due to the advantages of a clear mechanical concept, simple form and easy setup, the spring cell is widely used to simulate the bond slip between concrete and reinforcement. Therefore, this study simulates the bond slip through setting up spring units between the steel bars and the concrete. The direction perpendicular to the steel bars can be approximated as a rigid connection because the relative slip is small, so a linear spring cell can be chosen to establish. Since the elastic deformation in this direction is small, a larger stiffness factor is required to ensure the accuracy of the numerical simulation. In the direction along the length of the steel bars, the interaction between the steel bars and concrete will show obvious nonlinear characteristics because of the large relative slip, so the nonlinear spring element needs to be established to simulate its mechanical behavior. In this direction, the relationship between the force and displacement of the spring element is nonlinear. The bond-relative slip relationship for the nonlinear spring element is shown in Equation (15), and for uncorroded steel bars, the relationship is determined according to the local bond stress-slip relationship between uncorroded steel bars and concrete proposed by [PERSON] and [PERSON] [24], as shown in Equation (16):  
\[F=\tau(s)\cdot\tau d\l \tag{15}\]  
\[\tau(s)=(5.3\times 10^{2}s-2.52\times 10^{4}s^{2}+5.86\times 10^{5}s^{3}-5.4 7\times 10^{6}s^{4})\sqrt{\frac{f_{\mathrm{c}}}{40.7}} \tag{16}\]  
where \(F\) is the bond force of the spring element. \(l\) is the distance between adjacent spring elements. \(d\) is the diameter of the steel bar. \(\tau(s)\) is the bond stress between steel bars and concrete. \(f_{\mathrm{c}}\) is the compressive strength of concrete. \(s\) is the relative slip between steel bars and concrete.