## 4 Prediction of Reinforcement Tensile Force
### Response Surface Model of \(K_{t}\)  
Although it is difficult to obtain an explicit formulation of \(K_{\text{t}}\), one can do so using the response surface method. The basic idea of the response surface method is to assume a function that includes unknown parameters to replace the implicit actual expression. For example, given the ambiguous functional relationship \(Z\) = f(\(X_{1}\), \(X_{2}\),..., \(X_{\text{n}}\)) between the variable \(Z\) and the variables \(X_{1}\), \(X_{2}\),..., \(X_{\text{n}}\), the response surface method helps to establish an approximate function \(Z^{\prime}\) = f'(\(X_{1}\), \(X_{2}\),..., \(X_{\text{n}}\)) instead of the actual function through regressing and fitting data from a limited number of experiments. We used the response surface approximation proposed by [PERSON] (1970):  
\[y=\beta_{0}+\sum_{i=1}^{m}\beta_{i}x_{i}+\sum_{i=1}^{m}\beta_{ii}x_{i}^{2}+\sum _{i=1}^{m-1}\sum_{j=i+1}^{m}\beta_{ij}x_{i}x_{j} \tag{7}\]  
where \(y\) = the response value, \(\beta_{0}\), \(\beta_{i}\), \(\beta_{ii}\) and \(\beta_{ii}\) = the unsolved coefficients, and \(x_{i}\) and \(x_{i}\) = variables. When \(y\), \(x_{i}\), and \(x_{i}\) are known, \(\beta_{0}\), \(\beta_{i}\), \(\beta_{ii}\), and \(\beta_{ij}\) can be determined using the least-squares method. In the RSM of \(K_{\text{t}}\), \(K_{\text{t}}\) is regarded as the response value, and the active  
Figure 17: Reinforcement connection loads in baseline model with different toe interface friction angles. earth pressure coefficient \(K_{\rm a}\), toe interface friction angle \(\varphi_{\rm t}\), wall height \(H\), facing batter \(\omega\), reinforcement stiffness \(J\), reinforcement spacing \(S\), backfill friction angle \(\varphi\), and depth of reinforcement below the wall top \(z\) are regarded as variables. To reduce the number of and normalize the variables, the six above-mentioned factors can be formalized into the three variables as follows:  
\[x_{1}=\frac{z}{H} \tag{8}\]  
\[x_{2}=\frac{K_{a}}{\tan\varphi_{t}}=\frac{\cos^{2}(\phi+\omega)}{\cos^{3}w \cdot\left(1+\frac{\sin\phi}{\cos\omega}\right)^{2}\cdot\tan\varphi_{t}} \tag{9}\]  
\[x_{3}=\frac{S\cdot P}{J} \tag{10}\]  
where the influences of \(\omega\) and \(\varphi\) are considered through \(K_{\rm a}\) and \(P\) = the normal atmospheric pressure, which is used to normalize the reinforcement stiffness \(J\). Consequently, the RSM of \(K_{\rm t}\) can be written as the following:  
\[ \begin{array}{l}K_{\rm t}=\beta_{0}+\beta_{1}\big{(}\frac{z}{H}\big{)}+\beta _{2}\big{(}\frac{K_{\rm a}}{\tan\varphi_{\rm t}}\big{)}+\beta_{3}\big{(}\frac {S\cdot P}{J}\big{)}\\ +\beta_{11}\big{(}\frac{z}{H}\big{)}^{2}+\beta_{22}\big{(}\frac{K_{\rm a}}{ \tan\varphi_{\rm t}}\big{)}^{2}+\beta_{33}\big{(}\frac{S\cdot P}{J}\big{)}^{2 }\\ +\beta_{12}\big{(}\frac{z}{H}\big{)}\Big{(}\frac{K_{\rm a}}{\tan\varphi_{\rm t }}\big{)}+\beta_{23}\big{(}\frac{K_{\rm a}}{\tan\varphi_{\rm t}}\big{)}\Big{(} \frac{S\cdot P}{J}\big{)}+\beta_{13}\big{(}\frac{z}{H}\big{)}\Big{(}\frac{S \cdot P}{J}\big{)}\end{array}  \tag{11}\]  
A total of 705 values for the connection loads in different reinforcement layers were computed from the 65 numerical GRS walls mentioned previously and were used to establish a database of \(T_{\rm con,i}\). A corresponding database of \(K_{\rm t}\) can further be set up using Equation (6). To do so, substitute the values of \(K_{\rm t}\) and the corresponding values of \(z\), \(H\), \(K_{\rm a}\), \(S\), \(J\), and \(\varphi_{\rm t}\) into Equation (11). Then, the values of the unknown coefficients \(\beta_{0}\), \(\beta_{\rm i}\), \(\beta_{\rm ii}\), and \(\beta_{\rm ii}\) in Equation (11) can be determined using the least-squares method. Table 7 shows the values of \(\beta_{0}\), \(\beta_{\rm i}\), \(\beta_{\rm ii}\), and \(\beta_{\rm ii}\).