## 2 Methods
### Group Transfer Functions for Pile Shafts and Pile Bases
The relationship between the \"local shear displacement\" and the shaft shear stress as defined in Equation (2), was verified by laboratory test results conducted by ring shear tests on the soil-steel interface conducted by [PERSON] [23]. Detailed information about this field test case and the corresponding back-analysis method are described in later sections. Considering the pile-to-pile interaction, the soil displacement, \(W_{isz}\), for pile element \(i\) is because of shaft shear stress, \(\tau_{z}\), that the loading caused by its own pile load and additional interactive soil displacement is applied to the other piles in a pile group. Using the principle of superposition, the vertical soil-displacement of the pile element \(i\) at depth \(z\), \(W_{isz}\), consists of two parts: it is dependent upon the shaft shear stresses of pile element \(i\) at depth \(z\), \(W_{isz}\), and it is dependent on the interactive displacement effect, which is dependent on the adjacent pile \(j\), \(W_{ijz}\). Usually, the vertical displacement is obtained by calculating the formula proposed by [PERSON] and [PERSON] [7]. For a group of piles consisting of \(n\) piles, the displacement of the pile element \(i\) at depth \(z\) because of its own loading and adjacent pile action can be calculated using the following:  
\[W_{isz}=\frac{\tau_{o}}{G}\sum_{j=1}^{n}\tau_{jz}\ln(\frac{r_{m}}{s_{ij}}), \tag{5}\]  
[PERSON] and [PERSON] [21] proposed a simplification of this equation in that they assume that the shaft shear stress, \(\tau_{jz}\), at depth, \(z\), as expressed in Equation (5), is the same for all piles in a pile group (i.e., \(\tau_{iz}=\tau_{jz}\) for \(l=1\) to \(n\)). They verified the correctness and rationality of this hypothesis. Thus, the hypothetical constant values of \(\tau_{jz}\) (\(j=1\) to \(n\)) can cause Equation (5) to be simplified as follows:  
\[W_{isz}=\frac{\tau_{iz}\tau_{o}}{G}\sum_{j=1}^{n}\ln(\frac{r_{m}}{s_{ij}})= \overline{c}\tau_{iz}, \tag{6}\]  
where \(\overline{c}\) is a group parameter, which can be written as:  
\[\overline{c}=\frac{r_{o}}{G}\sum_{j=1}^{n}\ln(\frac{r_{m}}{s_{ij}}), \tag{7}\]  
Figure 2: Assumed shear stress and relative displacement relationship at the pile–soil interface. Thus, the load-transfer function of the pile groups will also be computed by substituting Equations (6) and (2) into Equation (1). Therefore, the total shaft displacement of pile \(i\) at depth \(z\) can also be written in the following form:  
\[s_{iz}=\frac{a\tau_{iz}}{1-b\tau_{iz}}+\overline{c}\tau_{iz}, \tag{8}\]  
Equation (8) can also be replaced with the following equation:  
\[\tau_{iz}=\frac{\Delta s_{iz}}{a+b\Delta s_{iz}}, \tag{2}\]0  
Equation (8) or Equation (9) is the load-transfer function of the pile shaft of the pile groups. The equation can also be presented in the following form:  
\[\tau_{iz}=\frac{\Delta s_{iz}}{a+b\Delta s_{iz}}, \tag{2}\]1  
where \(\widetilde{c}=\begin{array}{c}\sum\limits_{ \begin{array}{c}j=1\\ i\
eq j\end{array} }^{n}\frac{\left(1-v_{i}\right)}{2\Omega_{b}s_{ij}},\)\(\widetilde{a}\) is the inverse of the initial elastic soil stiffness of the pile base; thus, \(k_{i}^{b}=\frac{1}{\widetilde{a}}=\frac{4G\tau_{b}}{\left(1-v_{ib}\right)}\) and \(\widetilde{b}=\frac{p_{ib}R_{f}^{b}}{p_{bf}}\), where \(p_{ib}\) is the basic load of the movement, and \(G_{b}\) and \(v_{b}\) are the shear modulus and Poisson's ratio of the soil at the bottom of pile \(j\). \(R_{f}^{b}\) is the hyperbolic fit constant and \(p_{bf}\) is the ultimate base load (Chow [8]). Equation (10) can be written as:  
\[p_{ib}=\frac{\widetilde{a}+\widetilde{c}+\overline{b}w_{ib}-\sqrt{\left( \widetilde{a}+\widetilde{c}+\overline{b}w_{ib}\right)^{2}-4\overline{b} \widetilde{c}w_{ib}}}{2\widetilde{b}\widetilde{c}}, \tag{11}\]  
Equation (10) or Equation (11) is the load-transfer function at the pile base for a pile group analysis.