## 2 Finite Element Modeling and Validation
### Simulation Results and Correctness Verification  
To verify the reliability of the above finite element model, the strain cloud and load-midspan deflection curves of RC beams obtained from the finite element analysis are compared with the corresponding tests in this section, which are shown in Figures 6 and 7, respectively. The comparisons of the characteristic load and midspan deflection values of the strengthened beams with the specific values of the experimental results are shown in Table 3.  
In the CDP model, the development of cracks in RC beams is represented by tensile strains, and the tensile strain clouds of four specimens are shown in Figure 6. It can be seen from the figure that the crack distribution of each model is in good agreement with the test, and the plastic tensile strain of the unstrengthened beam B0 is greater and concentrated in the middle of the span, which is consistent with the test results that show that the cracks are concentrated in the middle of the span. Compared with the unstrengthened beam B0, the tensile strain of the FRP grid-PCM-composite-reinforced beam is reduced from 0.02287 to 0.02066, indicating that the maximum crack width of the concrete beam is significantly reduced, and the external FRP grid-PCM composite layer can effectively inhibit the formation and expansion of the cracks in reinforced concrete beams. Figure 5: Finite element model. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Beam ID**} & \multicolumn{2}{c}{**Yield Load (kN)**} & \multirow{2}{*}{**Error**} & \multicolumn{2}{c}{**Ultimate Load (kN)**} & \multicolumn{2}{c}{**Deflection Value (mm)**} & \multirow{2}{*}{**Error**} \\ \cline{2-3} \cline{5-10}  & **Experimental Value** & & & & **Simulation Value** & & & **(\%)** & **Experimental Value** & **Simulation Value** & **(\%)** \\ \hline B0 & 120.3 & 129.8 & 7.89 & 139.6 & 139.8 & \(-\)0.14 & 39.15 & 39.84 & 1.76 \\ BG-L-PCM & 159.2 & 164.6 & 3.39 & 223.9 & 211.8 & \(-\)5.40 & 22.00 & 21.14 & \(-\)3.90 \\ BG-M-PCM & 160.2 & 165.5 & 3.31 & 228.4 & 213.3 & \(-\)6.61 & 18.48 & 15.90 & \(-\)13.96 \\ BG-H-PCM & 163.2 & 166.3 & 1.90 & 219.5 & 217.1 & \(-\)1.09 & 13.85 & 14.63 & 5.63 \\ \hline \hline \end{tabular} Note: Error value = (simulated value – experimental value)/experimental value \(\times\) 100%; experimental values in the table are from the literature [15].  
\end{table} 
Table 3: Comparison of simulation results and experimental results. In the CDP model, the crack development of reinforced concrete beams is represented by equivalent plasticity (PEEQ), and the equivalent plastic strain cloud diagrams of the four specimens are shown in Figure 6. It can be seen from the figures that the crack distribution of each model is consistent with the test results, and the plastic tensile strain of the unreinforced beam B0 is greater and concentrated in the middle of the span, which is consistent with the test results. Compared with the unstrengthened beam B0, the equivalent strain of the FRP grid-PCM-composite-strengthened beam decreases from the top to bottom. The crack distribution of each model is consistent with the test results, and the plastic tensile strain of the unreinforced beam B0 is greater and concentrated in the middle of the span, which is consistent with the test results. Compared with the unstrengthened beam B0, the equivalent strain of the FRP grid-PCM-composite-strengthened beam decreases from the top to bottom. The crack distribution of each model is consistent with the test results, and the plastic tensile strain of the unreinforced beam B0 is greater and concentrated in the middle of the span, which is consistent with the test results. Compared with the unstrengthened beam B0, the equivalent strain of the FRP grid-PCM-composite-strengthened beam decreases from the top to bottom. The crack distribution of each model is consistent with the test results, and the plastic tensile strain of the unreinforced beam B0 is greater and concentrated in the middle of the span, which is consistent with the test results.