# Global Buckling Investigation of the Flanged Cruciform H-shapes Columns (FCHCs)
## 2 Calculation of Geometric Characteristics of the FCHC  
Theoretically, the cross-sections of the three hot-rolled H-shapes that make up the FCHC can be completely different. That means designers can choose three different cross-sections or different steel grades of hot-rolled H-beams for combination. However, from the simple design, the frame column section should be biaxially symmetrical or uniaxially symmetrical. Therefore, the other two H-shapes whose flanges are welded to another H-shape are the same section, as shown in Figure 2.  
Figure 1: The typical of built-up columns with H-shape: (**a**) UC or UB; (**b**) FCC; (**c**) BWS; (**d**) SS; (**e**) BBC; (**f**) FCHC.  
The letters C and SC stand for the centroid and shearing center of the H-shape, respectively, as shown in Figure 2, establishing three Cartesian coordinate systems. The whole section coordinate system, XOY, and the local x\({}_{1}\)O\({}_{1}\)y\({}_{1}\) coordinate system are set at H-shape-1 centroid (C1), the local x\({}_{2}\)O\({}_{2}\)y\({}_{2}\) coordinate system is established at the first H-shape-2 centroid (C2), and the local x\({}_{3}\)O\({}_{3}\)y\({}_{3}\) coordinate system is installed at the second H-shape-2 centroid (C3). For the H-shape, the section depth (H), the flange width (b\({}_{\rm f}\)), the flange thickness (t\({}_{\rm f}\)), and the web thickness (t\({}_{\rm w}\)) are also shown in Figure 2.  
The subscripts 1 and 2, respectively, represent the H-shape-1 and the H-shape-2. Therefore, under the premise of ignoring filled welds, in the XOY (x\({}_{1}\)O\({}_{1}\)y\({}_{1}\)) coordinate system, the \(X\)-axis coordinate value of the origin point O\({}_{2}\) (X\({}_{0}\),0) and O\({}_{3}\) (\(-X_{0}\),0) can be calculated as:  
\[X_{0}=\frac{H_{2}}{2}+\frac{t_{\rm w1}}{2}, \tag{1}\]  
where \(H_{2}\) is the depth of the H-shape-2; \(t_{\rm w1}\) is the web thickness of the H-shape-1.  
In the XOY coordinate system, the geometric characteristics of the combined section can be calculated according to the following equations.  
\[A=A_{1}+2A_{2}, \tag{2}\]  
\[I_{\rm X}=I_{\rm x1}+I_{\rm y2}+I_{\rm y3}=I_{\rm x1}+2I_{\rm y2}, \tag{3}\]  
\[I_{\rm Y}=I_{\rm y1}+2\cdot(I_{\rm x2}+A_{2}\cdot(X_{0})^{2}), \tag{4}\]  
\[i_{\rm X}=\sqrt{I_{\rm X}/A}, \tag{5}\]  
\[i_{\rm Y}=\sqrt{I_{\rm Y}/A}, \tag{6}\]  
\[r_{0}^{2}=i_{\rm X}^{2}+i_{\rm Y}^{2} \tag{7}\]  
\[J=\sum_{i=1}^{n}\frac{1}{3}b_{i}t_{i}^{3} \tag{8}\]  
\[C_{\rm w}=0 \tag{9}\]  
\(A\) is the cross-sectional area of the combined cross-section, and \(A=A_{1}+A_{2}\), \(A_{1}\) and \(A_{2}\) is the cross-sectional area of H-shape-1 and the H-shape-2, respectively.  
\(I_{\rm X}\) is the moment of inertia of the combined cross-section about the \(X\)-axis under the XOY coordinate system; \(I_{\rm Y}\) is the moment of inertia of the combined cross-section about the \(Y\)-axis under the XOY coordinate system.  
\(i_{\rm X}\) is the radius of gyration of the combined cross-section about the \(X\)-axis; \(i_{\rm Y}\) is the radius of gyration of the combined cross-section about the \(Y\)-axis.  
Figure 2: Geometric coordinates and dimensions of the FCHC.  
\(I_{\rm x1}\) and \(I_{\rm y1}\) is the moment of inertia of the H-shape-1 about the \(x_{1}\)-axis and \(y_{1}\)-axis, respectively, under the \(x_{1}\)O\({}_{1}y_{1}\) coordinate system.  
\(I_{\rm x2}\) and \(I_{\rm y2}\) is the moment of inertia of the H-shape-2 about the \(x_{2}\)-axis and \(y_{2}\)-axis, respectively, under the \(x_{2}\)O\({}_{2}\)Y\({}_{2}\) coordinate system.  
\(r_{o}\) is the polar radius of gyration about the shear center.  
\(J\) is the uniform torsional constant.  
\(C_{\rm w}\) is the warping constant.