# Force Analysis of Masonry Cave-Dwelling Structure Based on Elastic Center Method
## 3 Results and Analysis
### Comparative Analysis of Calculation Results of the Two Models  
Compared with the three-hinged arch, under the same load, the vertical reaction force V of the unhinged arch with the same arch axis is the same, and the horizontal thrust H is extremely close, while the thrust of the unhinged arch is larger, with the difference value of \((36.27-33.45)/33.45=8\%<10\%\). The bending moment at the three-hinged arch support is 0, while the restraint moment at the unhinged arch support is 6.68 kN\(\cdot\)m.  
The internal forces of the unhinged arch and the three-hinged arch at the section of the vault and the arch foot are calculated by Formulae (25)-(29), (35) and (36); they are shown in Table 1. It can be seen that the shear force and axial force of the unhinged arch model are not much different from the three-hinge arch model except for the bending moment in the section of the vault and the arch foot.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Calculation Model**} & \multirow{2}{*}{**Section Position**} & \multicolumn{2}{c}{**Bending Moment**} & \multirow{2}{*}{**Shearing Force Q (kN)**} & \multirow{2}{*}{**Axial Force**} \\  & & \(M\) (kN\(\cdot\)m) & & & **N (kN)** \\ \hline \multirow{2}{*}{unhinged arch} & arch vault & 2.44 & 0 & 36.27 \\  & arch foot & 6.68 & 24.31 & 68.59 \\ \multirow{2}{*}{three-hinged arch} & arch vault & 0 & 0 & 33.45 \\  & arch foot & 0 & 21.52 & 68.09 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Internal forces at the section of the vault and the arch foot under the two calculation models.  
Figure 10: Section internal force of right arch ring according to internal force formula of the unhinged arch and three-hinged arch: (**a**) bending moment \(M\); (**b**) shearing force \(Q\); and (**c**) axial force \(N\).  
It can be seen from Figure 10a that the bending moment of the arch section calculated by the unhinged arch model has two types: internal tension and external tension. There is a reverse bending point on the arch axis, and the maximum external tensile moment reaches 2.87 kN\(\cdot\)m, which appears at about one-third of the distance from the arch foot, while the internal maximum tensile moment occurs at the arch foot, reaching 6.68 kN\(\cdot\)m. The section bending moment calculated by the three-hinged arch model is only external tension, and the maximum bending moment is 7.41 kN m, which also appears at approximately one-third of the position from the arch foot. However, the absolute value of the maximum bending moment of the section calculated by the two mechanical models is approximately the same, and the difference is only (7.41 \(-\) 6.68)/7.41 = 9% \(<\) 10%. According to Figure 10b,c, the shear force and axial force calculated by the two models of the unhinged arch and three-hinged arch are basically the same as the changing trend of the section position; that is, they both gradually increase from the vault to the arch foot. The maximum shear forces of the two models both occur at the support and the values are not much different; the maximum shear force of the section calculated by the unhinged arch model is 24.31 kN, and that of the three-hinged arch model is 21.52 kN, which is slightly smaller than that of the unhinged arch model. The maximum axial forces of the two models occur near the support and the values were close. The maximum axial force of the section calculated by the unhinged arch model was 68.70 kN, and the maximum axial force of the section calculated by the three-hinged arch model was 68.11 kN; the axial force calculated by the unhinged arch was slightly larger. Therefore, it can be concluded that the maximum value of the bending moment, shear force, and axial force calculated by the two mechanical models are relatively close, and the selection of any mechanical model has little influence on the section design and recheck of the arch ring section.