## 5 Flexural Bearing Capacity of Concrete-Filled Steel Tubes
### Rectangular Composite Sections  
For the rectangular composite section, the plastic section modulus for the concrete core results from  
\[W_{c}=\frac{(B-2t_{w})\Big{(}H-2t_{f}\Big{)}^{2}}{4} \tag{19}\]  
and the plastic section modulus for the steel tube is given by  
\[W_{s}=\frac{BH^{2}-(\text{B}-2t_{w})\Big{(}\text{H}-2t_{f}\Big{)}^{2}}{4} \tag{20}\]  
The contribution of the concrete core to the maximum pure bending moment can be found by  
\[M_{c}=W_{c}f_{c}=\Bigg{[}\frac{(\text{B}-2t_{w})\Big{(}\text{H}-2t_{f}\Big{)}^ {2}}{4}\Bigg{]}t_{c} \tag{21}\]  
while the corresponding contribution of steel tube is given by  
\[M_{s}=W_{s}f_{y}=\Bigg{[}\frac{BH^{2}-(\text{B}-2t_{w})\Big{(}\text{H}-2t_{f} \Big{)}^{2}}{4}\Bigg{]}f_{y} \tag{22}\]  
To quantify the composite action for the aforementioned contribution of the discrete materials, the following empirical expressions can be expressed for the concrete core  
\[M^{\prime}_{c}=a_{c}M_{c}=\Bigg{[}a_{1}+a_{2}\Bigg{(}\frac{B}{t_{f}}+\frac{H} {t_{w}}\Bigg{)}+a_{3}\Bigg{(}\frac{f_{y}}{f_{c}}\Bigg{)}\Bigg{]}M_{c} \tag{23}\]  
Figure 6: Maximum axial force for circular CFTs; comparison of predicted and experimental results.  
where \(a_{c}\) and \(a_{1}\)-\(a_{3}\) are parameters that are presented in Table 5. Similarly, the effective axial force for the steel tube is given by  
\[M^{\prime}s=b_{s}M_{s}=\left(\text{b}_{1}+\text{b}_{2}\Bigg{(}\frac{B}{t_{f}}+ \frac{H}{t_{w}}\Bigg{)}+\text{b}_{3}\bigg{(}\frac{f_{y}}{f_{c}}\bigg{)}\right)M _{s} \tag{24}\]  
where \(\text{b}_{s}\) and \(\text{b}_{1}\)-\(\text{b}_{3}\) are parameters that are presented in Table 5. 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, as shown in Figure 1. Thus, according to this study, the maximum pure bending can be estimated by  
\[M_{prnd}=\text{c}\big{[}M^{\prime}c+M^{\prime}s\big{]} \tag{25}\]  
where parameter \(c\) takes into account the effect of composite column length and can be computed by  
\[\text{c}=\left(1-\text{c}_{1}\bigg{(}\frac{H}{L}\bigg{)}^{c_{2}}\right) \tag{26}\]  
where \(c_{1}\) and \(c_{2}\) are parameters that are presented in Table 5. Therefore, from Equations (23)-(26), one has  
\[M_{prnd}=\Bigg{[}\bigg{(}\text{a}_{1}+\text{a}_{2}\Bigg{(}\frac{B}{t_{f}}+ \frac{H}{t_{w}}\Bigg{)}+\text{a}_{3}\bigg{(}\frac{f_{y}}{f_{c}}\bigg{)}\Bigg{)} M_{c}+\Bigg{(}\text{b}_{1}+\text{b}_{2}\Bigg{(}\frac{B}{t_{f}}+\frac{H}{t_{w}} \Bigg{)}+\text{b}_{3}\bigg{(}\frac{f_{y}}{f_{c}}\bigg{)}\Bigg{)}M_{s}\Bigg{]} \bigg{(}1-\text{c}_{1}\bigg{(}\frac{H}{L}\bigg{)}^{c_{2}}\Bigg{)} \tag{27}\]  
Table 6 presents data from the experimental works of various researchers. Furthermore, the evaluation of the maximum bending moment using Equation (27) is also examined. It is found that the proposed empirical Equation (27) can predict the flexural capacity in a simple, yet effective and accurate manner. This is also obvious in Figure 7, where the predicted values of the maximum bending moment are close to the experimental ones.  
Figure 7: Maximum bending moment for rectangular CFTs; comparison of predicted and experimental results.  
\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
0.610416 & 0.005127 & 1.768951 & 12.923533 & \(-\)0.181790 & \(-\)0.079599 & 0.878259 & 0.006304 \\ \hline \end{tabular}
\end{table}
Table 5: Parameters for the prediction of the maximum bending moment–rectangular columns.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline