## 4 Axial Load-Bearing Capacity of Concrete-Filled Steel Tubes  
### Rectangular Composite Sections  
For the rectangular composite section, the area for the concrete core can be computed from  
\[A_{c}=(B-2t_{w})\Big{(}H-2t_{f}\Big{)} \tag{1}\]  
and the area for the steel tube is given by  
\[A_{s}=BH-(B-2t_{w})\Big{(}H-2t_{f}\Big{)} \tag{2}\]  
The strict contribution of the concrete core to the maximum axial load can be found by  
\[P_{c}=\mathrm{A_{c}}t_{c}=\Big{[}(\mathrm{B}-2t_{w})\Big{(}\mathrm{H}-2t_{f} \Big{)}\Big{]}t_{c} \tag{3}\]  
while the corresponding contribution of steel tube is given by  
\[P_{s}=\mathrm{A_{s}}t_{y}=\Big{[}BH-(\mathrm{B}-2t_{w})\Big{(}\mathrm{H}-2t_{ f}\Big{)}\Big{]}t_{y} \tag{4}\]  
To quantify the composite action for the aforementioned contribution of the discrete materials, the following empirical expressions can be expressed for the concrete core  
\[P^{\prime}_{c}=a_{c}P_{c}=\Bigg{[}a_{1}+a_{2}\Bigg{(}\frac{B}{t_{f}}+\frac{H}{ t_{w}}\Bigg{)}+a_{3}\Bigg{(}\frac{f_{y}}{f_{c}}\Bigg{)}\Bigg{]}P_{c} \tag{5}\]  
where \(a_{c}\) and \(a_{1}\)-\(a_{3}\) are parameters that are presented in Table 1. Similarly, the effective axial force for the steel tube is given by  
Figure 4: Comparison of experimental results (Ref. [18]) with the examined finite element models for concrete-filled steel tubes under pure bending.  
Figure 3: Comparison of experimental results (Ref. [17]) with the examined finite element models for concrete-filled steel tubes under pure bending.  
\[P^{\prime}{}_{s}=b_{s}P_{s}=\left(\text{b}_{1}+\text{b}_{2}\bigg{(}\frac{B}{t_{f}}+ \frac{H}{t_{w}}\bigg{)}+\text{b}_{3}\bigg{(}\frac{f_{y}}{f_{c}}\bigg{)}\right)P_{s} \tag{6}\]  
where \(\text{b}_{s}\) and \(\text{b}_{1}\)-\(\text{b}_{3}\) are parameters that are presented in Table 1. It should be mentioned that B and H are the rectangular section's width and height, respectively, and \(\text{t}_{f}\) and \(\text{t}_{w}\) are the steel tube section thicknesses for the flange and web, respectively. Thus, according to this study, the predicted strength for axial load results from  
\[P_{pred}=c\big{[}P^{\prime}{}_{c}+P^{\prime}{}_{s}\big{]} \tag{7}\]  
where parameter \(c\) takes into account the effect of composite column length and can be  
\[\text{c}=\bigg{(}1-\text{c}_{1}\bigg{(}\frac{H}{L}\bigg{)}^{\text{c}_{2}}\bigg{)} \tag{8}\]  
where \(c_{1}\) and \(c_{2}\) are the parameters that are presented in Table 1. Therefore, from Equations (5)-(8), one has  
\[P_{pred}=\Bigg{[}\bigg{(}a_{1}+a_{2}\bigg{(}\frac{B}{t_{f}}+\frac{H}{t_{w}} \bigg{)}+a_{3}\bigg{(}\frac{f_{y}}{f_{c}}\bigg{)}\bigg{)}P_{c}+\Bigg{(}b_{1}+b _{2}\bigg{(}\frac{B}{t_{f}}+\frac{H}{t_{w}}\bigg{)}+b_{3}\bigg{(}\frac{f_{y}}{ f_{c}}\bigg{)}\bigg{)}P_{s}\Bigg{]}\bigg{(}1-c_{1}\bigg{(}\frac{H}{L}\bigg{)}^{ \text{c}_{2}}\bigg{)} \tag{9}\]  
Table 2 presents data from the experimental works of various researchers. Furthermore, the evaluation of maximum axial load using Equation (9) is also examined. It is found that the proposed empirical Equation (9) can predict the axial load capacity in a simple, yet effective and accurate manner. This is also obvious in Figure 5, where the predicted axial forces are close to the experimental ones.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \(a_{1}\) & \(a_{2}\) & \(a_{3}\) & \(b_{1}\) & \(b_{2}\) & \(b_{3}\) & \(c_{1}\) & \(c_{2}\) \\ \hline
1.01837 & 0.002135 & 0.032575 & 1.882731 & \(-\)0.00397 & \(-\)0.01295 & 0.41722 & 0.038095 \\ \hline \end{tabular}
\end{table}
Table 1: Parameters for the prediction of axial load–rectangular columns.  
Figure 5: Maximum axial force for rectangular CFTs; comparison of predicted and experimental results.