# Bending Performance of Concrete Sandwich Walls with Actual Boundary Conditions
## 5 Analytical Model of Composite Action  
The degree of composite action (\(K\)) of the sandwich wall is an important factor that can reveal the interactive working degree of the components. When \(K\) equals to 0, it means that the two layers of concrete worked separately and the moment of inertia of the wall should be the summation of two layers of concrete. When \(K\) equals to 1, it means that the two layers of concrete works together, the moment of inertia of the wall should be calculated based on the whole section. Currently, there are two methods assessing the composite action of sandwich walls. One is related to the level of stiffness [25], and the other is related to the bearing capacity [26], both of which are expressed as:  
\[K_{\mathrm{s}}=\frac{I_{0}-I_{\mathrm{nc}}}{I_{\mathrm{c}}-I_{\mathrm{nc}}} \tag{1}\]  
\[K_{b}=\frac{F_{0}-F_{\mathrm{nc}}}{F_{0}-F_{\mathrm{nc}}} \tag{2}\]  
in which \(K_{\mathrm{s}}\) and \(K_{\mathrm{b}}\) denote the degree of composite action related to stiffness and bearing capacity, respectively. \(I_{\mathrm{nc}}\) and \(I_{\mathrm{c}}\) represent the moment of inertia of the sandwich wall, with the degrees of composite action being 0 and 1, respectively. \(I_{0}\) is the moment of inertia of the sandwich wall when reaching the cracking load. \(F_{\mathrm{nc}}\) and \(F_{\mathrm{c}}\) represent the ultimate bearing capacity of the sandwich wall with composite action degrees of 0 and 1, respectively. \(F_{0}\) represents the ultimate bearing capacity of the sandwich wall. Figure 15: Stress distributions of specimen WPO. (**a**) Stress of reinforcement wire. (**b**) Stress of steel truss parallel to the span. (**c**) Stress of steel truss perpendicular to the span. (**d**) Stress of strengthened steel truss. The non-English term ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ means ‘Average’. Generally, the degree of composite action for the sandwich wall gradually decreased with the increase in load. Figure 16 presents the strain distribution of the sandwich wall with \(K\) values of 0 and 1. For the sandwich wall before cracking, the moment of inertia could be expressed as [27]:  
\[A_{0}=bh+(n-1)A_{\mathrm{s}} \tag{3}\]  
\[x_{0}=\frac{0.5 bh^{2}+(n-1)A_{\mathrm{s}}h_{0}}{bh+(n-1)A_{\mathrm{s}}} \tag{4}\]  
\[I_{0}=\frac{b}{3}\Big{[}x_{0}^{3}+(h-x_{0})^{3}\Big{]}+(n-1)A_{\mathrm{s}}(h_{ 0}-x_{0})^{2} \tag{5}\]  
in which \(A_{\mathrm{s}}\) is the area of reinforcements under tension, and \(b\) and \(h\) represent the width and thickness of the sandwich wall. After cracking, the moment of inertia could be expressed as:  
\[x_{\mathrm{cr}}=(\sqrt{n^{2}\mu^{2}+2n\mu}-n\mu)h_{0} \tag{6}\]  
\[I_{\mathrm{cr}}=\frac{1}{3}bx_{\mathrm{cr}}^{3}+nA_{\mathrm{s}}(h_{0}-x_{ \mathrm{cr}})^{2} \tag{7}\]  
in which \(\mu=A_{\mathrm{s}}/bh_{0}\) and \(n=E_{\mathrm{s}}/E_{0}\). The ultimate bearing capacity could be easily calculated according to the strain distribution. Tables 3 and 4 list the degree of composite action for test specimens based on their strength and stiffness, respectively. It could be observed that the degree of composite action of the test specimens based on their bearing capacities ranged from 33.05% to 45.53%, and, based on the moment of inertia, varied from 58.65% to 66.83%. It could also be seen that the degree of composite action of the sandwich walls under negative direction was slightly higher than that under