## Fiber Beam Element Model of A CFST Member with the Strain Rate Effect  
### Material Constitutive Models  
#### 2.1.1 Constitutive Model of Steel  
The bilinear hardening skeleton curve (Figure 1) recommended in the Chinese Code for Design of Concrete Structures GB 50010-2010 [40] is selected as the stress-strain skeleton curve of steel and is shown in the following formula.  
\[\sigma=\left\{\begin{array}{ccc}E\varepsilon&\varepsilon\leq\varepsilon_{y} \\ f_{y}+k_{1}(\varepsilon-\varepsilon_{y})&\varepsilon_{y}<\varepsilon<\varepsilon _{u}\\ 0&\varepsilon\geq\varepsilon_{u}\end{array}\right. \tag{1}\]  
where \(E_{s}\) is [PERSON]'s modulus of steel, \(\varepsilon_{y}\) and \(f_{y}\) are the strain and stress at the yielding point, respectively, \(\varepsilon_{u}\) is the strain at the ultimate stress, and \(k_{1}\) is the slope of the hardening branch.  
The loading and unloading rules of steel proposed by [PERSON] [41] shown in Figure 2 are used, which can effectively consider the stiffness degradation during the loading and unloading process. In this study, the rules are appropriately simplified and corrected to address the numerical stability problem and the inherent deficiency in the model. In particular, the unloading is considered to be an approximately linear elastic process, the corresponding calculation is based on the initial elastic modulus \(E_{s}\), and the stress associated with the reverse reloading (shown as 1 and 2 in Figure 2) is calculated using the following equation.  
\[\sigma=E_{Z}(\varepsilon-\varepsilon_{a,i}) \tag{2}\]  
Figure 1: The stress-strain skeleton curve of steel.  
\[E_{Z}=\frac{\sigma_{b,i}-\sigma_{a,i}}{\varepsilon_{b,i}-\varepsilon_{a,i}} \tag{3}\]  
where \(E_{z}\) is the reloading stiffness, \(\sigma_{a,i}\) and \(\varepsilon_{a,i}\) are the respective stress and strain at the \(i\)th reverse loading point, and \(\sigma_{b,i}\) and \(\varepsilon_{b,i}\) are the respective maximum stress and strain point in the last loading step. If the steel has never yielded in the reloading direction, the loading path points to the initial yield strain and stress. If the steel has yielded in the reloading direction, the loading path points to the historical maximum strain point.  
The conditions where reloading occurs after incomplete unloading in the skeleton curve (shown as 3 in Figure 2)--that is, where the strain values at the loading point and the unloading point are very close--should be properly handled. A numerical accuracy issue may lead to an unstable value for the reloading stiffness \(E_{z}\). Essentially, \(E_{z}\) can be larger than \(E_{s}\), which is inconsistent with the actual conditions and is prone to generating program operation errors. In this case, the stress should be approximately calculated using \(E_{s}\). As shown in Figure 3, if unloading occurs at point A (using the initial stiffness \(E_{s}\)), then the cyclic loading and unloading occur between two relatively close strains, and the unloading stiffness is higher than the loading stiffness. Thus, the stress value decreases, which does not reflect the actual conditions. Therefore, for incomplete unloading in the reloading curve (4 in Figures 2 and 3), both loading and unloading stresses are calculated using the initial stiffness. After reaching the stress value of the previous unloading point, the loading and unloading stresses are calculated using the actual stiffness.  
Figure 3: The stress error of steel under cyclic loading.  
Figure 2: The cyclic constitutive model of steel.