## 2 Numerical Finite Element Model  
### Outline  
The numerical model is outlined in Figure 1. It uses a circular section without stiffeners in the vertical direction, with a diaphragm placed directly above the concrete inside the steel tube. The diaphragm has the effect of smoothly transferring compressive forces to the concrete and preventing the progression of buckling even after the maximum bearing capacity. \(L\), \(L_{\text{c}}\), \(D\), and \(t\) in the figure indicate the height of the steel tube, the height of the concrete, and the diameter and thickness of the steel tube, respectively. The specifications of each numerical model are shown in Table 1. The radius thickness ratio parameter \(R\) and the slenderness ratio parameter \(\lambda\) in the table are expressed by the following equations by Specifications for highway bridges in Japan [9]:  
\[R=\sqrt{3(1-\mu_{\text{s}}^{2})}\frac{f_{\text{s}}}{E_{\text{s}}}\frac{D}{2t} \tag{1}\]  
\[\lambda=\frac{KL}{r}\frac{1}{\pi}\sqrt{\frac{f_{\text{s}}}{E_{\text{s}}}} \tag{2}\]  
where \(\mu_{\text{s}}\), \(f_{\text{s}}\), \(E_{\text{s}}\), \(K\), and \(r\) are Poisson's ratio, yield stress, Young's modulus, the effective buckling length factor of the cantilevered beam, and the secondary radius of the section of steel tube, respectively. Based on the parameter limits defined in the Japanese Specifications for Highway Bridges [9], \(R\) (\(0.03\leq R\leq 0.12\)) is set to 0.06, 0.09, and 0.12, and \(\lambda\) (\(0.2\leq\lambda\leq 0.4\)) is set to 0.2, 0.3, and 0.4. The axial compressive force acting on the column head ranged from 10% to 50% of the yield axial force of the steel tube only, excluding the in-filled concrete. Figure 1: Numerical model. The stress-strain curve of the steel is shown in Figure 2. Young's modulus \(E_{\text{s}}\), [PERSON]'s ratio \(\mu_{\text{s}}\), and yield stress \(f_{\text{s}}\) were set to 206 GPa, 0.3, and 308 MPa, respectively, based on material tests obtained from a report on cyclic load testing of concrete-filled circular section bridge piers conducted at the Public Works Research Institute [10]. The von [PERSON] yield condition was adopted as the yield criterion. For the concrete, the uniaxial compressive strength \(f_{\text{c}}\) was set to 22.0 MPa, [PERSON]'s modulus \(E_{\text{c}}=4730\sqrt{t_{\text{c}}}=21.0\) GPa, and [PERSON]'s ratio \(\mu_{\text{c}}\) was set to 0.2 according to ACI Committee 318 [11]. For the concrete, the stress-strain curve (Figure 3) was referred to the Popovics model and made into a multilinear form [12]. For the yield condition, the Drucker-Prager failure criterion was used as expressed in the following equation:  
\[f_{\text{c}}=\alpha I_{1}+\sqrt{I_{2}} \tag{3}\] \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Model** & **C-006-020** & **C-006-030** & **C-006-040** & **C-009-020** & **C-009-030** & **C-009-040** & **C-012-020** & **C-012-030** & **C-012-040** \\ \hline Diameter of steel tube & & & & & & & & & \\ \(D\) [mm] & & & & & & & & & \\ Thickness of steel tube & & & & & & & & & \\ \(t_{\text{s}}\) [mm] & & 18 & & & 12 & & & 9.3 & \\ Height of steel tube & 2550 & 3850 & 5100 & 2600 & 3850 & 5100 & 2600 & 3850 & 5100 \\ \(L\) [mm] & & 0.06 & & & 0.09 & & & 0.12 & \\ Radius thickness ratio & & & & & & & & & \\ parameter \(R\) & & & & & & & & & \\ Slenderness ratio & 0.20 & 0.30 & 0.40 & 0.20 & 0.30 & 0.40 & 0.20 & 0.30 & 0.40 \\ \(\lambda\) & Axial force ratio \(n\) & & & & 0.1, 0.2, 0.3, 0.4, 0.5 & & & & \\ Concrete filling ratio \(L_{\text{c}}/L\) & & & & & 0.2, 0.4, 0.6 & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Numerical model specifications. Figure 3: Compressive stress–compressive strain curve for concrete filling. Figure 2: Steel tube stress-strain curve. Here, \(I_{1}\) is the first invariant of the stress tensor, \(J_{2}\) is the second invariant of the deviatoric stress tensor, and the material constant \(\alpha\) is assumed to be 0.2 based on [PERSON]'s triaxial compression test [13].