# Global Buckling Investigation of the Flanged Cruciform H-shapes Columns (FCHCs)
## 5 Results, Observations, and Discussion
### Global Flexural Buckling Factors in Current Design Codes  
In GB50017-2017 [16], the global flexural buckling factor \(\varphi\) of the axially loaded compression member should be calculated according to the following formula. When \(\lambda_{\text{n}}\leq 0.215\),  
\[\varphi=1-\alpha_{1}\lambda_{\text{n}}^{2}, \tag{10}\]  
When \(\lambda_{\text{n}}>0.215\),  
\[\varphi=\frac{1}{2\lambda_{\text{n}}^{2}}\bigg{[}\left(\alpha_{2}+\alpha_{3} \lambda_{\text{n}}+\lambda_{\text{n}}^{2}\right)-\sqrt{\left(\alpha_{2}+\alpha _{3}\lambda_{\text{n}}+\lambda_{\text{n}}^{2}\right)^{2}-4\lambda_{\text{n}}^ {2}}\bigg{]}, \tag{11}\]  
where \(\lambda_{\text{n}}\) is dimensionless slenderness ratio, \(\lambda_{\text{n}}=\frac{\lambda}{\pi}\sqrt{f_{y}/E}\). \(\alpha_{1}\), \(\alpha 2\), and \(\alpha_{3}\) is the coefficient, classified according to the section classification and adapted according to Table 5.  
In EN 1993-1-1 [17], the global flexural buckling curve of the axially loaded compression member is expressed according to the following. \[\varphi=\frac{1}{\Phi+\sqrt{\Phi^{2}-\lambda_{\text{n}}^{2}}}\text{ but }\varphi\leq 1.0, \tag{12}\]  
where, \(\Phi=0.5[1+\alpha(\lambda_{\text{n}}-0.2)+\lambda_{\text{n}}^{2}]\), \(\alpha\) is an imperfection factor corresponding to the appropriate buckling curve should be obtained from Table 6.  
Figure 10: The FCHC-2-7 series illustrates the typical failure mode of the flexural buckling: (**a**) FCHC-2-7-3; (**b**) FCHC-2-7-6; (**c**) FCHC-2-7-9; (**d**) FCHC-2-7-12. ANSI/AISC 360-16 [18] only has one column buckling strength curve, expressed according to the following. When \(\lambda_{\rm n}\leq 1.5\),  
\[\varphi=0.658\frac{F_{\rm y}}{\rm{e}}, \tag{13}\]  
When \(\lambda_{\rm n}>1.5\),  
\[\varphi=\frac{0.877F_{\rm e}}{F_{\rm y}}, \tag{14}\]  
\(F_{\rm e}=\frac{\pi^{2}E}{\lambda^{2}}\), \(E\) is the modulus of elasticity of steel; \(F_{\rm y}\) is the specified minimum yield stress of the type of steel being used (MPa). The buckling capacity (the peak load \(N_{\rm u}\)), global flexural buckling factor \(\varphi\) (\(\varphi=N_{u}/(f_{y}\cdot A)\)), slenderness ratio \(\lambda\) (\(\lambda=L_{\rm c}/i\)), and dimensionless slenderness ratio \(\lambda_{\rm n}\) (\(\lambda_{\rm n}=\frac{\lambda}{\pi}\sqrt{f_{\rm y}/E}\)) were obtained and summarized in Tables 7 and 8. It should be noted that, in Tables 7 and 8, the letters b and c of Curve b and Curve c represent the column buckling curves of type of b and c according to cross-section classifications taken from GB50017 and EN1993-1-1, respectively. For all Q345 specimens, the ABAQUS results (\(\varphi^{\rm A}\)) are lower than the corresponding design curve (i.e., curve b) in both GB 50,017 (Figure 11a) and EN 1993-1-1 (Figure 11b) by an average of \(-1.2\) and \(-1.4\%\), respectively, and higher than the curve c in GB 50,017 by 11.5% and EN 1993-1-1 by 5.9%. The FE results (\(\varphi^{\rm A}\)) are also lower than the design curve in ANSI/AISC 360 (Figure 11c) by \(-7.5\%\) on average. However, for the Q460 specimen, the \(\varphi^{\rm A}\) is lower than the corresponding design curve b in both GB 50,017 (Figure 12a) and EN 1993-1-1 (Figure 12b) by an average of \(-0.9\) and \(-0.8\%\), respectively, and higher than the curve c in GB 50,017 by 11.8% and EN 1993-1-1 by 6.8%. In addition, the \(\varphi^{\rm A}\) is lower than the design curve in ANSI/AISC 360 (Figure 12c) by \(-7.5\%\) on average. Thus, the design provision in ANSI/AISC 360-16 [18] is unsuitable for the global buckling design of FCHCs. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Column Buckling Curve** & \(\mathbf{a_{0}}\) & \(\mathbf{a}\) & \(\mathbf{b}\) & \(\mathbf{c}\) & \(\mathbf{d}\) \\ \hline Imperfection factor \(\alpha\) & 0.13 & 0.21 & 0.34 & 0.49 & 0.76 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Imperfection factors for buckling curves in EN 1993-1-1 [17]. Figure 11: Comparison of FE results with design curves for Q345 specimens: (**a**) GB50017; (**b**) EN1993-1-1; (**c**) ANSI/AISC360.