## 3 Analysis of the Flexural Performances of CCFSts with Gaps
### Numerical Model Configuration  
In the numerical model of CCFSTs, the concrete core is simulated by 3D 8-node reduced integral elements. The outer steel tube is simulated by the 4-node reduced integral shell element. By referring to the mesh in a study by [PERSON] et al. (2016) [9], this study sets a denser mesh to ensure the reliability of the numerical results. The concrete damage plastic (CDP) constitutive model is adopted [51,52,53]. The Poisson's ratio, \(\mu\), cylinder compressive strength,\(f_{c}\ell^{{}^{\prime}}\), and corresponding strain, \(\varepsilon_{c}\ell^{{}^{\prime}}\), of the concrete under uniaxial compressive stress are 0.21, 33 MPa, and 0.0033, respectively. The empirical Equation (10), recommended in ACI318 (2011), can be adopted to calculate the elastic modulus, \(E_{c}\)[54]. \[E_{c}=4700\sqrt{f_{c}^{\prime}} \tag{10}\]  
The ratio of the compressive strength under biaxial loading to uniaxial compressive strength,\(f_{b0}/f_{c}^{\prime}\), can be calculated using Equation (11) [55]. \[\frac{f_{b0}}{f_{c}^{\prime}}=1.5\big{(}f_{c}^{\prime}\big{)}^{-0.075} \tag{11}\]  
where \(f_{b0}\) is the biaxial compressive strength. The confinement factor, \(\xi\), can be determined according to Equation (12) [3]. \[\xi=\frac{A_{s}f_{s}}{A_{c}f_{c}^{\prime}} \tag{12}\]  
According to [PERSON] et al. [56], the ratio of the second stress invariant on the tensile meridian to that of the compressive meridian, \(K_{c}\), and the dilation angle, \(\psi\), can be determined by Equations (13) and (14). \[K_{c}=\frac{5.5}{5+2(f_{c}^{\prime})^{0.075}} \tag{13}\]  
\[\psi=\left\{ \begin{array}{l}56.3(1-\xi),\xi\leq 0.5\\ 6.672e^{\frac{7.4}{164+\xi+\xi}},\xi>0.5\end{array} \right. \tag{14}\]  
According to Equations (10)-(14), the key material parameters for confining the CDP constitutive model are set. Namely, the elastic modulus, \(E_{c}\), is 27 GPa; the ratio of the second stress invariant on the tensile meridian to that on the compressive meridian, \(K_{c}\), is 0.72. Moreover, the shear dilation angle depends on the confining pressure parameters. Because of the different gap sizes set in this study, the shear dilation angle and confinement factor vary with the changes in the gap size conditions. The flow potential eccentricity, \(e\), is 0.1. The viscosity parameter is set to \(1\times 10^{-5}\) to ensure that the model has appropriate convergence. When core concrete has to bear a circular confining pressure, the uniaxial compressive yield strength, \(f_{c}^{\prime}\), and corresponding strain, \(\varepsilon_{c}^{\prime}\), are substantially higher than those of unconfined concrete. According to [PERSON]'s model, the relationship between \(f_{c}^{\prime}\) and its corresponding strain, \(\varepsilon_{c}^{\prime}\), as well as the parameters (\(f_{cc}^{\prime}\) and \(\varepsilon_{cc}^{\prime}\)) under confining compression [57; 58], can be defined. [PERSON] et al. [9] established an equivalent stress-strain relationship to simulate the plastic-damage behavior of core concrete in CCFSts under a compressive state, as follows:  
\[\overline{\sigma}=\left\{ \begin{array}{l}2\overline{\varepsilon}-\overline{ \varepsilon}^{2},\overline{\varepsilon}\leq 1\\ \left\{\begin{array}{l}1+q\cdot\left(\overline{\varepsilon}^{0.1\xi}-1\right),\xi\geq 1.12\\ \frac{\overline{\varepsilon}}{\left(\beta\cdot\left(\overline{\varepsilon}-1 \right)^{2}+\overline{\varepsilon}\right)},\xi<1.12\end{array}\right\}, \overline{\varepsilon}>1\end{array} \right. \tag{15}\]  
\[\sigma_{0}=\left(1+\left(-0.054\xi^{2}+0.4\xi\right)\left(\frac{24}{f_{c}^{ \prime}}\right)^{0.45}\right)f_{c}^{\prime} \tag{16}\]  
\[\varepsilon_{0}=\varepsilon_{cc}+\left(1400+800\left(\frac{f_{c}^{\prime}}{2 4}-1\right)\right)\xi^{0.2} \tag{17}\]  
\[q=\frac{\xi^{0.745}}{2+\xi} \tag{18}\]  
\[\beta=2.35\times 10^{-4}\left(2.36\times 10^{-5}\right)^{\left(0.25+\left( \xi-0.5\right)^{7}\right)}f_{c}^{\prime 2} \tag{19}\]  
Additionally, the tension stiffening still needs to be defined in ABAQUS.