## Fiber Beam Element Model of A CFST Member with the Strain Rate Effect
### Strain Rate Effect
#### 2.2.1 Strain Rate Effect of Steel  
Currently, there are few studies on the influence of the strain rate on the nonlinear analysis of structures, and the constitutive model of metal plastic material in ABAQUS can consider the impact of the strain rate only on the yield strength or ultimate strength. To address this issue, a subroutine for steel that can effectively consider the effect of strain rate is developed in this study. Existing studies have shown that the yield strength, tensile strength, ultimate strain, and slope of the hardening branch of the stress-strain curve of steel are all influenced by the strain rate. In addition, when the strain rate changes during loading, the shape of the hysteresis loop changes to a certain extent, and the area of the hysteresis loop increases with a rising strain rate [9,41]. Considering only the influence of the strain rate on the yield strength and, in particular, modifying only the strength using the dynamic increase factor method cannot accurately describe the influence of the time-varying strain rate on the mechanical properties of steel and ignores some important factors relatively sensitive to the strain rate. To address this problem, the following issues are considered in this study. The influences of the strain rate on the yield strength and tensile strength are considered in real time. According to the actual needs, different dynamic constitutive models can be selected for the strain rate effect of steel. In this study, the dynamic tensile constitutive model of steel provided in Reference [17] is selected, as shown in the following formulas. \[\frac{f_{yd}}{f_{ys}}=1+c_{f}\lg\frac{\dot{\varepsilon}}{\dot{\varepsilon}_{0}} \tag{12}\]  
\[\frac{f_{ud}}{f_{us}}=1+c_{u}\lg\frac{\dot{\varepsilon}}{\dot{\varepsilon}_{0}} \tag{13}\]  
\[c_{f}=0.1709-3.289\times 10^{-4}f_{ys} \tag{14}\]  
\[c_{u}=0.02738-2.982\times 10^{-5}f_{ys} \tag{15}\]  
where \(\dot{\varepsilon}\) is the strain rate, \(\dot{\varepsilon}_{0}\) denotes the quasi-static strain rate, \(f_{ys}\) and \(f_{yd}\) are the quasi-static and dynamic yield strength, respectively, and \(f_{us}\) and \(f_{yd}\) are the quasi-static and dynamic ultimate tensile strength, respectively. The influence on the slope of the hardening branch of the steel stress-strain curve due to strain rate is directly considered by substituting Equations (12)-(14) into those representing the skeleton curve. The influence of the strain rate on the area of the hysteresis loop is considered by correcting the historical maximum strain and stress. The strain of steel consists of elastic strain \(\varepsilon_{e}\) and plastic strain \(\varepsilon_{p}\). Therefore, the historical maximum strain \(\varepsilon_{un}\) at a certain strain rate is equal to the sum of the historical maximum plastic strain in this direction and the elastic strain \(\sigma_{un}/E_{s}\) under this strain rate. The historical maximum stress \(\sigma_{un}\) is taken as \(f_{y}+\Delta\sigma\), where \(\Delta\sigma\) is the product of the historical maximum plastic strain and the slope of the hardening branch under a quasi-static state. Additionally, the influence of the strain rate on the elastic modulus, plastic strain, and strain rate history are ignored because no consistent conclusions were obtained in previous studies. When a structure is subjected to dynamic loading (e.g., seismic load) with a varying loading rate in real time, the loading path of steel may repeatedly switch between elastic loading and elastoplastic loading. The stress of the steel dramatically varies in real time, which necessitates the consideration of the strain rate's influence on the elastic ultimate strain. To ensure the stability of the numerical results and the accuracy of the constitutive model, the following considerations are examined.