# Stressing State Analysis on a Single Tube CFST Arch Under Spatial Loads
## 4 Stressing State Analysis of the Arch Model
### Modelling of Stressing State Submodes  
As for the internal forces of the arch model, it withstands axial force, bending moment, and torsion under spatial loads. Due to the little effect of torsion, it can be ignored in analysis [14]. Thus, the stressing state mode of the arch can be mainly decomposed into stressing state submodes for axial compression and the bending moment, respectively, to characterize their working behavior and illustrate their different roles in the loading process. Longitudinal strains measured around each cross section are the most direct reflection of axial force. However, the strains of separate points have high variability and low determinacy, so they could not accurately judge the stressing state submode for axial compression. Therefore, the mean value of sectional strains is calculated as a quantitative index, called the \"axial strain\" \(\varepsilon_{N}\), to represent the action effect of axial compression at the sectional centroid point, that is  
\[\varepsilon_{N}=\frac{1}{4}(\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3} +\varepsilon_{4}). \tag{10}\]  
Thus, the \(\varepsilon_{N}\) of all these sections can be calculated to embody the developing trend of the stressing state submode for the axial force. To better observe the increasing trend of \(\varepsilon_{N}\) between adjacent loads, the difference value \(\Delta\varepsilon_{N}\) is also calculated. As shown in Figure 5, the stressing state submode for axial forces and the increasing trend of some typical sections are plotted. It shows that the curves are divided into three different developing trends. Before 66 kN, the \(\varepsilon_{N}\) increases slowly with no obvious mutation; during 66 kN to 85 kN, the \(\varepsilon_{N}\) of some sections increase rapidly; after 85 kN, the curves signify the unstable stressing state. Due to the sensitivity of the arch foot, the \(\varepsilon_{N}\) shows great values at Section 13. Besides, the axial forces in the arch appear as compression under normal circumstances, so the \(\varepsilon_{N}\) should be negative normally. But it is found in Figure 5a that the \(\varepsilon_{N}\) present positive at Section 1, while other sections are all negative. This may be caused by an inaccuracy in the measurement or by the geometric and material nonlinearity of the arch, whereby local buckling of steel tube may also be produced. Compared to the submode for axial force, the stressing state submodes for bending moments of the arch model come into greater focus. Similar to the construction of \(\varepsilon_{N}\), sectional strains are applied to construct the parameter \"generalized bending strain\" \(\varepsilon_{M}\), to quantitatively describe the submodes for bending moments. Under spatial loads, the bending moments can be decomposed into in-plane bending moments and out-plane ones. Hence, the strains \(\varepsilon_{1}\), \(\varepsilon_{2}\) and \(\varepsilon_{3}\), \(\varepsilon_{4}\) are divided to calculate the out-plane \(\varepsilon_{M\text{-out}}\) and in-plane \(\varepsilon_{M\text{-in}}\), respectively, they are  
\[\varepsilon_{M\text{-out}}=\frac{1}{2}(\varepsilon_{1}-\varepsilon_{2}),\ \varepsilon_{M\text{-in}}=\frac{1}{2}(\varepsilon_{3}-\varepsilon_{4}). \tag{11}\]  
Thus, the \(\varepsilon_{M}\) of all these sections are calculated to embody the distribution of stressing state submodes for in-plane or out-plane bending moments. To better observe their growth trend between adjacent loads, the differences \(\Delta\varepsilon_{M\text{-out}}\) or \(\Delta\varepsilon_{M\text{-in}}\) are also calculated. Figure 5: The stressing state submodes for axial force: (**a**) \(\varepsilon_{N}\)-\(F\) curves; (**b**) \(\Delta\varepsilon_{N}\)-\(F\) curves. Figure 6 shows the changing features of the arch model's stressing state submodes for bending moments, as well as the corresponding characteristic parameters \(\Delta\varepsilon_{M\text{-out}}\) and \(\Delta\varepsilon_{M\text{-in}}\).