## 5 Results and Analysis  
### Model Validation  
In order to verify the validity of the numerical model, the comparison between the numerically calculated and experimentally obtained bending moment-rotational angle curve is conducted. The schematic diagrams of the calculation of the moment and rotational angle are illustrated in Figure 6 and they can be calculated using Equations (1) and (2), respectively.  
\[M=(F+G)\times l-G\times l_{1}-F\times l_{2}-N\times e \tag{1}\]  
where \(M\) is bending moment of the joint section, \(F\) is the vertical load, \(G\) is the self-weight of the segment, \(N\) is the horizontal load, and \(e\) is the vertical eccentric distance of \(N\). The \(l\), \(l_{1}\), and \(l_{2}\) are the horizontal distances of the three forces from the joint section, respectively.  
\[\theta=2 arctan\frac{\delta_{1}-\delta_{2}}{2h} \tag{2}\]  
where, \(\theta\) is the rotational angle (unit: radians), \(\delta_{1}\) is the joint-opening change at the internal surface (unit: mm), \(\delta_{2}\) is the joint-closing change at the external surface (unit: mm), and \(h\) is the thickness of the segment, which is 350 mm.  
Figure 5: Schematic diagram of virtual-tracking-element technology: (**a**) the initial model; (**b**) creation of VE; (**c**) removal of OE; (**d**) deformation of VE; (**e**) activation of OE; (**f**) continue of loading.  
Figure 6: Schematic diagram of the calculation of: (**a**) bending moment; (**b**) rotational angle.  
The corresponding comparison between the numerical and experimental results of the unreinforced and reinforced specimens are shown in Figures 7 and 8, respectively. It can be seen that both numerical curves are in good agreement with the experimental results, and that the curves can be divided into a couple of sections.  
As demonstrated in Figure 7, in the OA section, the specimen is in the elastic stage, with low deformation and high stiffness. This result was slightly larger than the test result due to the manufacturing errors in the experiment. The deformation accelerated in the AB section, since the bent bolts began to yield. In the BC section, a temporary increase in the stiffness occurred, since the external concrete was squeezed and in close contact, and the plastic zone started to form; in the ultimate-failure stage (CD section), the compression stress exceeded the bearing capacity of the concrete and led to failure (the plastic zone penetrated through the concrete), and the ultimate moment was 193 kN\(\cdot\)m.  
Figure 8: Comparison between numerical and experimental results of the reinforced specimen.  
Figure 7: Comparison between numerical and experimental results of the unreinforced specimen.  
Regarding the reinforced specimen (Figure 8), the point a is the reinforcement point, corresponding to the elastic limit (point a in Figure 7), and the stiffness in the OA section is the same as the unreinforced specimen, indicating that the virtual tracking element had no effect on the specimen. The AB section manifested similar stiffness from the unreinforced specimen, which was attributed to the interspace between the punched-hole wall of the SSCP and the chemical anchor, making SSCP unable to share the load. When the chemical anchor made contact with the hole wall (point b), the stiffness had a notable increase due to the high reinforcement capability of the SSCP. Therefore, the increase in stiffness often lagged behind the reinforcement point. In the CD section, the stiffness gradually decreased because the trough of the SSCP entered the yielding stage, which undermined its capacity to resist deformation. In the final stage, the failure mode was the same as that in the unreinforced specimen (concrete cracked), and the ultimate moment was 464 kN\(\cdot\)m, which increased by 140.4% compared to the unreinforced specimen.