## 5 Flexural Bearing Capacity of Concrete-Filled Steel Tubes
### Circular Composite Sections  
For the circular composite section, the plastic section modulus for the concrete core results from  
\[\text{W}_{c}=\frac{\left(\text{D}-2t\right)^{3}}{6} \tag{28}\]  
and the plastic section modulus for the steel tube is given by  
\[\text{W}_{s}=\frac{D^{3}-\left(\text{D}-2t\right)^{3}}{6} \tag{29}\]  
The contribution of the concrete core to the maximum pure bending moment can be found by  
\[\text{M}_{c}=\text{W}_{c}t_{c}=\left[\frac{\left(\text{D}-2t\right)^{3}}{6} \right]\text{f}_{c} \tag{30}\]  
while the corresponding contribution of steel tube is given by  
\[\text{M}_{s}=\text{W}_{s}\text{f}_{y}=\left[\frac{D^{3}-\left(\text{D}-2t \right)^{3}}{6}\right]\text{f}_{y} \tag{31}\]  
To quantify the composite action for the aforementioned contribution of the discrete materials, the following empirical expressions can be expressed for the concrete core  
\[\text{M}^{\prime}_{c}=a_{c}\text{M}_{c}=\left[a_{1}+a_{2}\bigg{(}\frac{D}{t} \bigg{)}+a_{3}\bigg{(}\frac{f_{y}}{f_{c}}\bigg{)}\right]\text{M}_{c} \tag{32}\]  
where a\({}_{c}\) and a\({}_{1}\)-a\({}_{3}\) are parameters that are presented in Table 7. Similarly, the effective axial force for the steel tube is given by  
\[\text{M}^{\prime}_{s}=b_{s}\text{M}_{s}=\bigg{(}\text{b}_{1}+\text{b}_{2} \bigg{(}\frac{D}{t}\bigg{)}+\text{b}_{3}\bigg{(}\frac{f_{y}}{f_{c}}\bigg{)} \bigg{)}\text{M}_{s} \tag{33}\]  
where b\({}_{s}\) and b\({}_{1}\)-b\({}_{3}\) are parameters that are presented in Table 7. It should be mentioned that D and t are the steel tube's diameter and thickness, respectively, as shown in Figure 1. Thus, according to this study, the maximum pure bending can be estimated by  
\[\text{M}_{prat}=\text{c}\big{[}\text{M}^{\prime}_{c}+\text{M}^{\prime}_{s} \big{]} \tag{34}\]  
where parameter c takes into account the effect of composite column length and can be computed by  
\[\text{c}=\bigg{(}1-\text{c}_{1}\bigg{(}\frac{H}{L}\bigg{)}^{\text{c}_{2}} \bigg{)} \tag{35}\]  
where c\({}_{1}\) and c\({}_{2}\) are parameters that are presented in Table 7. Therefore, from Equations (23)-(26), one has  
\[\text{M}_{prat}=\bigg{[}\bigg{(}\text{a}_{1}+\text{a}_{2}\bigg{(}\frac{B}{t_ {f}}+\frac{H}{t_{w}}\bigg{)}+\text{a}_{3}\bigg{(}\frac{f_{y}}{f_{c}}\bigg{)} \bigg{)}\text{M}_{c}+\bigg{(}\text{b}_{1}+\text{b}_{2}\bigg{(}\frac{B}{t_{f}} +\frac{H}{t_{w}}\bigg{)}+\text{b}_{3}\bigg{(}\frac{f_{y}}{f_{c}}\bigg{)} \bigg{)}\text{M}_{s}\bigg{]}\bigg{(}1-\text{c}_{1}\bigg{(}\frac{H}{L}\bigg{)} ^{\text{c}_{2}}\bigg{)} \tag{36}\]  
Table 8 presents data from the experimental works from previous studies where the evaluation of the maximum bending moment using Equation (36) is also shown. It is found  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\text{a}_{1}\) & \(\text{a}_{2}\) & \(\text{a}_{3}\) & \(\text{b}_{1}\) & \(\text{b}_{2}\) & \(\text{b}_{3}\) & \(\text{c}_{1}\) & \(\text{c}_{2}\) \\ \hline
7.868303 & \(-0.05441\) & 271.8801 & 207.2656 & \(-45.2648\) & \(-0.30532\) & 0.96368 & 0.002373 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Parameters for the prediction of maximum bending moment–circular columns.