# Stressing State Analysis on a Single Tube CFST Arch Under Spatial Loads
## 4 Stressing State Analysis of the Arch Model  
### Investigation into the E-F Curve  
By calculating GSED values with strains of each measured point according to Equations (1)-(3), it can directly model the stressing state mode of the arch under spatial loads. Thus, the _E-F_ curve is plotted to describe the working characteristics of the arch model during the whole loading process, as shown in Figure 3a. To distinguish the first characteristic load in the curve, the M-K criterion is applied to achieve the intersecting point of the _UF_ and _UB_ curves based on the sequence of {\(E_{j,\text{norm}}\)} from 0 kN to 97 kN. As for the second characteristic load, the sequence {\(E_{j,\text{norm}}\)} is from 66 kN to 97 kN. Hence, two characteristic loads, \(P\) (66 kN) and \(Q\) (85 kN), are obtained. According to the two loads, the structural stressing state curve of the arch model is directly divided into three different stages. Before load \(P\), the curve presents steady changes with load increase, indicating that the arch is basically in the linear-elastic working state in this stage. After load \(P\), the arch model is in the elastic-plastic working state due to the local plastic development of steel tube. According to the natural law from quantitative change to qualitative change, characteristic load \(Q\) is detected, whereas thereafter a trend of relatively sharper increase of the \(E\) emerges and the arch enters the unstable working state until its finally failure at ultimate load (97 kN). Since the arch model is subjected to in-plane and out-plane loads simultaneously, the formation of strains \(\varepsilon_{1}\), \(\varepsilon_{2}\) and \(\varepsilon_{3}\), \(\varepsilon_{4}\) are considered the action results of both axial forces but out-plane and in-plane bending moments, respectively. Thus, the strains \(\varepsilon_{1}\), \(\varepsilon_{2}\) and \(\varepsilon_{3}\), \(\varepsilon_{4}\) of one section can be normalized as the out-plane and in-plane GSED values, respectively, to observe their stressing state characteristics. When calculating the value of out-plane GSEDs (\(E_{\text{out}}\)) by Equations (1)-(3), only \(\varepsilon_{1}\) and \(\varepsilon_{2}\) of each cross section are considered, similarly, we consider strains \(\varepsilon_{3}\), \(\varepsilon_{4}\) only to calculate the in-plane GSEDs (\(E_{\text{in}}\)). Therefore, the _E-F_ curves for out-plane and in-plane ones can be plotted, as illustrated in Figure 3b. From loads \(P\) to \(Q\), the \(E_{\text{in}}\)-_F_ curve has an obvious change at 75 kN, while the \(E_{\text{out}}\)-_F_ curve keeps a normal increase. At 85 kN, \(E_{\text{out}}\) has an obvious change (turning point) together with a little trend change of \(E_{\text{in}}\). It seems that the in-plane and out-plane working behaviors are relatively independent before load \(P\) (66 kN); but from loads \(P\) to \(Q\) (85 kN), the in-plane and out-plane working behaviors could be coupled with each other. After point \(Q\), the whole arch is in failure state controlled by out-plane behaviors. According to the investigation above, two common characteristics are summarized to describe the stressing state of the arch model. Firstly, characteristic load \(P\) is the branch point that the stressing state of the arch changes from elasticity to elastoplasticity. The whole structure keeps a stable stressing state until load \(Q\). In other words, the arch's stressing state keeps quantitative change rather than qualitative change. Secondly, characteristic load \(Q\) is the branch point where the arch's stress state qualitatively changes/leaps to an unstable stressing state state that is different from before. Hence, the load at point \(Q\) is defined as the failure load of the arch structure to represent the starting point of structural failure  
Figure 3: The generalized strain energy density-based curves of the arch model: (**a**) the _E-F_ curve and characteristic loads; (**b**) the \(E_{\text{in}}\)-_F_ and \(E_{\text{out}}\)-_F_ curves. process.