## 4 XFEM-Based Two-Scale Simulation on RC Columns under Cyclic Loading
### Relationship between Lateral Displacement and Reaction Force  
Under lateral cyclic loading, the relationship between lateral reaction force and horizontal displacement at the loading point of tested specimens simulated with the solid element model, two-scale model and that of test results is summarized in Figure 13a-c. The peak values of reaction forces are summarized in Table 3. It can be seen that the numerical simulation results are in good agreement with experimental results. The hysteresis loops corresponding to each loading cycle are consistent with that of tested specimens, presenting a remarkable pinch phenomenon. Figure 13d shows the comparison of the hysteretic behavior and the corresponding crack patterns of RC columns with different ACRs. From simulation results, four symmetrical cracks on both sides of the specimens C1 to C3 can be observed at the end of the cyclic loading. As shown in Figure 13d, the height of cracking position led by tension and compressive crushing area at the corner of column foot will be lower with the increment of the ACR, which is similar to experimental observation shown in Figure 10a-c.  
Figure 11: Time-history curves of steel strain. Figure 12: Time–history curves of crack width. Therefore, the cracking pattern in tested RC columns can be accurately forecast using the XFEM-based two-scale modeling approach proposed in this study. As listed in Table 3, the peak values of reaction forces in hysteresis curves obtained from the two-scale model are slightly higher than those of the solid element model. Certain errors exist between positive and negative peak values corresponding to the forward and reverse loadings. The fundamental reason for this phenomenon is that the fiber-beam element is incapable of simulating concrete cracking, and the reduction of material strength and stiffness can only be reflected according to the reduction of Young's modulus and material strength in constitutive law. Since concrete is heterogeneous material, microcracking forms even at the initial loading stage due to high stress status at local regions, which cannot be simulated with fiber elements. Compared with the high-precision solid elements, a certain degree of error is inevitable. The maximum principal stress criterion is introduced to the material constitutive model of concrete to determine the initiation and propagation of cracking in  
Figure 13: Comparison of load–displacement curves of different specimens. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Model Type**} & **Positive Peak** & **Positive Peak** & **Negative Peak** & **Negative Peak** \\  & **Value (kN)** & **Error (\%)** & **Value (kN)** & **Error (\%)** \\ \hline C1-experimental results & 84.15 & - & \(-\)87.66 & - \\ C1-Two-scale & 89.33 & 6.15 & \(-\)93.87 & 7.09 \\ C2-experimental results & 89.53 & - & \(-\)93.06 & - \\ C2-Two-scale & 93.09 & 3.98 & \(-\)97.33 & 4.58 \\ C3-experimental results & 106.43 & - & \(-\)102.33 & - \\ C3-Two-scale & 96.95 & \(-\)8.90 & \(-\)102.44 & 0.11 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of peak load values. XFEM elements. In ABAQUS, the asymptotic crack-tip function \(F_{a}(x)\) is neglected, which makes the crack-tip unable to stay inside the element and cut through at least one whole element once element stress reaches the maximum principal stress. Therefore, a slight difference in element stress may lead to the propagation paths slightly varying from each other. Moreover, it should be noted that the peak value errors of reaction force between numerical simulation results and experimental data are within 9%, as illustrated in Table 3. The peak loads of a two-scale model of specimen C1 are 6.15% and 7.09% higher than that of experimental results. For specimen C3, the positive load peak of the two-scale model is 8.90% smaller than that of experimental results.