# Axial and Bending Bearing Capacity of Double-Steel-Concrete Composite Shear Walls
## 4 Derivation of the Axial and Bending Bearing Capacity
### Large Eccentric Compression Failure (\(\xi\leq\xi_{b}\))  
Figure 8 shows the distribution of the cross-section stress and strain for the large eccentric SC wall. The equilibrium equation of the cross section is described as Equations (8) and (9).  
\[\sum X=0,\ N=A_{s}^{\prime}f_{y}+C_{c}+N_{sw}-A_{s}f_{y} \tag{8}\]  
\[\sum M=0,\ Ne=A_{s}^{\prime}f_{y}h_{w}+M_{c}+M_{sw} \tag{9}\]  
where, \(N\) is the design value of the axial force; \(e\) is the distance from the axial force to the center of the steel plate in tension; \(A_{s}\)' is the area of the steel plate in compression; \(N_{sw}\) is the resultant force of the web plate (stress in compressive direction is defined positive and stress in tensile direction is negative); \(M_{sw}\) is the moment of the web plate normal stress about the center of the steel plate in tension (the clockwise direction is defined as positive, the counterclockwise direction is defined as negative); \(C_{c}\) is the resultant of the concrete compressive stress (stress in compressive direction is defined positive and stress in tensile direction is negative); and \(M_{c}\) is the moment of the concrete normal stress about the center of the steel plate in tension (the clockwise direction is defined as positive, the counterclockwise direction is defined as negative).  
If \(\beta_{1}x_{n}>h_{f}^{\prime}\),  
\[C_{c}=\alpha_{1}f_{cc}\beta_{1}x_{n}b_{w}+\alpha_{1}f_{cc}\left(b_{f}^{\prime} -b_{w}\right)h_{f}^{\prime} \tag{10}\]  
\[M_{c}=\alpha_{1}f_{cc}\beta_{1}x_{n}b_{w}\left(h_{w}-\frac{\beta_{1}x_{n}}{2} \right)+\alpha_{1}f_{cc}\left(b_{f}^{\prime}-b_{w}\right)h_{f}^{\prime}\left( h_{w}-\frac{h_{f}^{\prime}}{2}\right) \tag{11}\]  
If \(\beta_{1}x_{n}\leq h_{f}^{\prime}\),  
\[C_{c}=\alpha_{1}f_{cc}\beta_{1}x_{n}b_{f}^{\prime} \tag{12}\]  
\[M_{c}=\alpha_{1}f_{cc}\beta_{1}x_{n}b_{f}^{\prime}\left(h_{w}-\frac{\beta_{1} x_{n}}{2}\right) \tag{13}\]  
where, \(x_{n}\) is the height of the concrete compressive zone; \(b_{w}\) is the thickness of the wall web; \(h_{f}^{\prime}\) is the height of the compression flange; and \(b_{f}^{{}^{\prime}}\) is the width of the compression flange.  
Figure 7: The strain distribution of the SC walls.  
The stress distribution of the web plate is equivalent to the superimposition of one full plastic stress distribution and a triangle stress distribution, as shown in Figure 9. The triangle stress distribution can be seen as a couple. The couple has a negligible effect on the bearing capacity. For example, the horizontal bearing capacity of SCW1-5 in [12] was 1653.57 kN if the couple was considered. The result was 1653.59 kN if the couple was ignored and the difference was only 0.0012%. Therefore, the couple in Figure 9b can be ignored in the process of calculating the bearing capacity. The formula is simplified as Equations (14) and (15).  
\[N_{sw}=f_{y}t_{sw}(2x_{n}-h_{w}) \tag{14}\]  
\[M_{sw}=f_{y}t_{sw}\bigg{[}\frac{1}{2}h_{w}^{2}-(h_{w}-x_{n})^{2}\bigg{]} \tag{15}\]  
Figure 8: Stress and strain distribution of cross section for the large eccentric SC wall. (**a**) Strain distribution of the cross section; (**b**) Stress distribution of the cross section.  
Figure 9: Equivalence of web steel plate for the large eccentric SC wall. (**a**) Full plastic stress distribution; (**b**)Triangle stress distribution.