## 5 Stressing State Analysis of CFST Arch Models
### Investigation into Structural Stressing State Submodes on Internal Forces  
The CFST arch models mainly underwent axial compression and bending moment. After constructing the strain/stress fields based on the NSF method, the distribution modes of sectional internal forces can also be obtained. Thus, the stressing state submodes of sectional axial forces and bending moments were built respectively to reflect their various working characteristics  
Figure 11: Stress fields of arch models around characteristic loads \(Q\): (**a**) Section 11 of arch-A; (**b**) Section 13 of arch-B. and evaluate their roles on arch models. Equations (10) and (11) were proposed to calculate sectional internal forces of axial force (\(N\)) and bending moment (\(M_{y}\)), respectively:  
\[N=\int_{A}\sigma\mathrm{d}A=\sum_{A}\sigma_{i}A_{i} \tag{10}\]  
\[M_{y}=\int_{A}\sigma\mathrm{d}A=\sum_{A}\sigma_{i}y_{i}A_{i} \tag{11}\]  
For arch-A, the out-plane bending moment was not analyzed, owing to the lateral restraint. Therefore, Figure 12 only shows the stressing state submodes of axial forces (\(N\)) and in-plane bending moments (\(M_{\mathrm{in}}\)). In Figure 12a, \(N\) at section 8 was smaller than other sections. Closer to the later part of load, this phenomenon became more pronounced. Combining the great displacements and \(M_{\mathrm{in}}\) at sections 7 and 9 indicates that the bending moment played a control role in the section. Before the load exceeded 140 kN, the distribution mode of \(N\) was stable with light undulation. Then, the distribution mode began to show obvious fluctuations at some sections, such as sections 2, 7, and 11. However, the mutation features around the failure load could not be clearly distinguished from the curve. In Figure 12b, it is seen that the curves were in asymmetrical distribution, especially under high-level loading. Excluding sections 7 and 9, which were mentioned previously, section 11 also had large bending moments, indicating that the area near the arch feet was subjected to heavy force. The developing trend of \(M_{\mathrm{in}}\) after 180 kN changed with great increments, especially at sections 7-9, which reflected the qualitative change of stressing submodes at the failure load. For arch-B, Figure 13 shows the stressing state submodes of axial forces (\(N\)), in-plane bending moments (\(M_{\mathrm{in}}\)), and out-plane bending moments (\(M_{\mathrm{out}}\)). Under normal circumstances, axial forces in arch appeared as compression, but \(N\) presented positive at section 1 while others were all negative. This may have been caused by an inaccurate measurement or geometric and material nonlinearity  
Figure 12: The stressing state submodes for internal forces of arch-A: (**a**) The distribution mode of \(N\); (**b**) the distribution mode of \(M_{\mathrm{in}}\). of the arch, and local buckling of steel tube may have also been produced. The same phenomenon occurred in the curve of \(M_{\text{out}}\). Nevertheless, this did not have a large effect on the trend features of the arch's stressing state. As shown in Figure 13a,b, the curves developed steadily with no obvious mutation changes, reflecting that the arch maintained excellent in-plane stability in all loading processes. As depicted in Figure 13c, except the \(M_{\text{out}}\) near the arch feet, the curves from sections 3-11 were symmetrically distributed around section 7 (the arch vault), demonstrating that the development of \(M_{\text{out}}\) was not sensitive to initial imperfections. Out-plane point load was applied at the arch vault, so the maximum \(M_{\text{out}}\) occurred at section 7. The distribution mode of \(M_{\text{out}}\) clearly showed great increments after 85 kN, which verifies that the updated failure load was reasonable. From the description of stressing state submodes of internal forces, their distribution modes and developing trends were reflected.