# Experimental and Numerical Studies on the Structural Performance of a Double Composite Wall
## 3 Numerical Approach  
### Non-Linear Flexural Analysis Model  
Figure 9 shows a non-linear flexural analysis model for evaluating the flexural behavior of the DCW member. Since no damage was observed at the interface between the PC panel and CIP concrete in the DCW-SN specimen, the interface layers of the PC panel and CIP concrete were assumed to be fully bonded in the flexural analysis for this member. Under the assumption of the concrete strain at the extreme compression fiber (\(\varepsilon_{t}\)) and the neutral axis depth (\(c\)), the strain in concrete (\(\varepsilon_{c,i}\)) at an arbitrary distance (\(d_{i}\)) away from the extreme compression fiber, the strain of tensile reinforcement (\(\varepsilon_{s}\)), and the strain of compressive reinforcement (\(\varepsilon_{s}^{\prime}\)) can be calculated as follows:  
\[\varepsilon_{c,i}=\varepsilon_{t}-\frac{\varepsilon_{t}}{c}d_{i}, \tag{4}\]  
\[\varepsilon_{s}=\varepsilon_{t}-\frac{\varepsilon_{t}}{c}d_{s}, \tag{5}\]  
\[\varepsilon^{\prime}{}_{s}=\varepsilon_{t}-\frac{\varepsilon_{t}}{c}d_{s}^{ \prime}. \tag{6}\]  
where \(d_{s}\) and \(d_{s}^{\prime}\) are the depths of the tensile and compressive reinforcements, respectively. The stresses corresponding to each strain can be calculated by substituting the strain of each element derived from Equations (4) to (6) into the constitutive laws of the materials. In this study, the Popovics model [17] was applied to concrete, while the elasto-linear hardening model [18] was applied to steel reinforcement, as shown in Figure 9a. \[f_{c}=-f_{c}^{\prime}\left[2\left(\frac{\varepsilon_{c}}{\varepsilon_{c}^{ \prime}}\right)-\left(\frac{\varepsilon_{c}}{\varepsilon_{c}^{\prime}} \right)^{2}\right]\left(\text{when }\varepsilon_{c}\leq 0\right), \tag{7}\]  
\[f_{s}=\varepsilon_{s}E_{s}\left(\text{when }\varepsilon_{s}\leq\varepsilon_{y} \right), \tag{8}\]  
\[f_{s}=f_{y}+\left(\varepsilon_{s}-\varepsilon_{y}\right)E_{sp}\left(\text{when } \varepsilon_{s}>\varepsilon_{y}\right), \tag{9}\]  
where \(f_{c}^{\prime}\) is the compressive strength of concrete; \(\varepsilon_{c}^{\prime}\) is the compressive strain of concrete corresponding to \(f_{c}^{\prime}\); \(E_{s}\) is the elastic modulus of steel reinforcement (190 GPa is used based on the material test); \(f_{y}\) and \(\varepsilon_{y}\) are the yield strength and strain of steel reinforcement, respectively; and \(E_{sp}\) is the elastic modulus of the steel reinforcement after yielding, which is taken to be \(0.01E_{s}\). Figure 8: Shear contribution of each element. When the concrete stress (\(f_{c,i}\)) of each layer is calculated using Equations (4) and (7), the compressive force of concrete (\(C_{c}\)) can be calculated by accumulating the stress of each layer, as follows:  
\[C_{c}=\sum_{\varepsilon_{i},i\leq 0}f_{c,i}b_{i}z_{i}, \tag{10}\]  
where \(b_{i}\) and \(z_{i}\) are the width and height of each concrete layer, respectively. The force of tensile reinforcement (\(T_{s}\)) and the force of compressive reinforcement (\(T_{s}^{\prime}\)) can be calculated by multiplying the stress estimated using Equations (8) and (9) by the sectional area of the steel reinforcement, and \(C_{c}\), \(T_{s}\), and \(T_{s}^{\prime}\) should meet the following force equilibrium condition:  
\[\sum F=C_{c}+T_{s}+T_{s}^{\prime}=0. \tag{11}\]  
Figure 9: Nonlinear flexural analysis model: (a) schematic description of flexural analysis; (b) numerical integration for obtaining displacement.