# Flexural Capacity of a New Composite Beam with Concrete-Infilled Tubular Lower Flange
## 3 Flexural Strength Estimation
### Non-Composite Specimens
\begin{table}
\begin{tabular}{c c} \hline
**Internal Axial Force Component** & **Force Equation** \\ \hline Compression force at upper flange (\(C_{Af}\)) & \(C_{Af}=b_{f}\times t_{f}\times F_{y}\) \\ Compression force at web above PNA (\(C_{Aw}\)) & \(C_{Aw}=t_{w}\times(y_{PNA}-t_{f})\times F_{y}\) \\ Tension force at web below PNA (\(T_{Aw}\)) & \(T_{Au}=t_{w}\times(h_{A}-y_{PNA})\times F_{y}\) \\ Tension force at upper flange of tube (\(T_{twf}\)) & \(T_{twf}=b_{u}\times t_{u}\times F_{y}\) \\ Tension force at web of tube (\(T_{tw}\)) & \(T_{tw}=2t_{u}\times(h_{u}-2t_{u})\times F_{y}\) \\ Tension force at lower flange of tube (\(T_{tbf}\)) & \(T_{tbf}=b_{u}\times t_{u}\times F_{y}\) \\ Tension force at main rebars (\(T_{re}\)) & \(T_{re}=n\times A_{re}\times F_{y,re}\) \\ \hline \end{tabular}
\end{table} 
Table 4: Internal force components in case the plastic neutral axis exists at the web. \begin{table}
\begin{tabular}{c c} \hline
**Internal Axial Force Component** & **Force Equation** \\ \hline Compression force at upper flange (\(C_{Af}\)) & \(C_{Af}=b_{f}\times t_{f}\times F_{y}\) \\ Compression force at web (\(C_{Aw}\)) & \(C_{Aw}=t_{w}\times(h_{A}-t_{f})\times F_{y}\) \\ Compression force at upper flange of tube above & \(C_{twf}=b_{u}\times(y_{PNA}-h_{A})\times F_{y}\) \\ Tension force at upper flange of tube below PNA & \(T_{twf}=b_{u}\times(h_{A}+t_{u}-y_{PNA})\times F_{y}\) \\ Tension force at web of tube (\(T_{tw}\)) & \(T_{tw}=2t_{u}\times(h_{u}-2t_{u})\times F_{y}\) \\ Tension force at lower flange of tube (\(T_{tbf}\)) & \(T_{tbf}=b_{u}\times t_{u}\times F_{y}\) \\ Tension force at main rebars (\(T_{re}\)) & \(T_{re}=n\times A_{re}\times F_{y,re}\) \\ \hline \end{tabular}
\end{table} 
Table 5: Internal force components in case the plastic neutral axis exists at the upper flange of the tube. In case the plastic neutral axis exists at the web of the tube  
The internal axial force for each component of the section can be computed as shown in Table 6. It is assumed that the contribution of all concrete components to the flexural strength of the member is ignored because only a small portion of concrete is subjected to compression force, as shown in Figure 9c. The location of the PNA for this case can be determined from the axial force equilibrium condition expressed below:  
\[C_{Af}+C_{Aw}+C_{tw}+C_{tw}=T_{tw}+T_{bbf}+T_{re}. \tag{6}\]  
By using the PNA location determined from the above equation, the plastic moment of the given composite section can be calculated by:  
\[ \begin{array}{ll}M_{p}&=C_{Af}\times(y_{PNA}-\frac{t_{f}}{2})+C_{Aw}\times(y _{PNA}-t_{f}-\frac{(h_{A}-t_{f})}{2})+C_{tw}\times(y_{PNA}-h_{A}-\frac{t_{w}}{2 })+C_{tw}\times(\frac{y_{PNA}-h_{A}-t_{u}}{2})\\ &+T_{tw}\times(\frac{H-y_{PNA}-t_{u}}{2})+T_{tbf}\times(H-y_{PNA}-\frac{t_{w}}{ 2})+T_{re}\times(H-y_{PNA}-h_{re})\end{array} . \tag{7}\]  
The lateral torsional moment of the non-composite beam sections can be calculated by following the procedure introduced in the LRFD specification. Since the composite beam proposed in this study has a different cross-sectional shape from the conventional wide flange sections, the warping constant (\(C_{w}\)) and torsional constant (\(J\)) need to be evaluated by the following equations:  
\[C_{w}=a^{2}I_{1}+b^{2}I_{2}, \tag{8}\]  
\[J=\frac{b_{f}t_{f}^{3}}{3}+\frac{4\times{A_{p}}^{2}}{p/t_{u}}, \tag{9}\]  
where several new variables in the above equations can be computed by  
\[ \begin{array}{ll}I_{1}=\frac{t_{f}\times b_{f}^{3}}{12},&I_{2}=2\times(\frac {t_{u}b_{f}^{3}}{12}+\frac{h_{u}t_{u}^{3}}{12}+t_{u}h_{u}\times(\frac{b_{u}}{ 2}-\frac{t_{u}}{2})^{2})+2\times(\frac{\pi\times{r_{w}}^{4}}{2}+\pi\times{r_{ w}}^{2}\times((\frac{b_{u}}{8})^{2}+(\frac{3\times b_{u}}{8})^{2})),\\ \\ \rho=\frac{I_{1}}{I_{1}+I_{2}},&a=(1-\rho)\times(H-\frac{t_{f}}{2}-\frac{h_{u} }{2}),&b=\rho\times(H-\frac{t_{u}}{2}-\frac{h_{u}}{2}),\\ \\ A_{p}=(b_{u}-t_{u})\times(h_{u}-t_{u}),&p=2\times((b_{u}-t_{u})+(h_{u}-t_{u})). \end{array}