## 2 Methods  
### Group Transfer Functions for Pile Shafts and Pile Bases  
Based on the nonlinear analysis method of the pile groups proposed by [PERSON] and [PERSON] [20; 21], the change law of the soil subjected to a disturbance decreases with the increase in the distance in the base shaft. It is assumed that any given depth of soil is the total displacement, \(s_{z}\), of the axial displacement along the pile that contains the nonlinear displacement of the destruction area and the elastic region of the soil (Figure 1a). At the same time, the nonlinear displacement is defined by [PERSON] and [PERSON] [21] as \"local shear displacement\", \(\Delta s_{z}\), the total displacement of the pile in the depth, \(z\), position, which is caused by the pile axial shear stress \(\tau_{z}\), and a narrow disturbance area is formed around the pile shaft. The stress-strain, compressibility and thickness changes in the narrow disturbed zone perch are complex and are mainly influenced by the construction method of the pile, the type of pile, the type of soil, and the loading procedure. In addition, the nonlinear soil displacement characteristics caused by the axial shear force of the pile can be approximated as noncontinuous (Figure 1b). It is assumed that the soil outside the \"local shear displacement\" is elastic, and the soil displacement caused by the axial shear stress, \(\tau_{z}\), of the pile is the elastic displacement, \(W_{sz}\). Thus, the shaft displacement, \(s_{zz}\), caused by pile element \(i\) in a pile group, and when given a depth, \(z\), of the soil, can be calculated using the following formula:  
\[s_{iz}=\Delta s_{iz}+W_{isz}, \tag{1}\]  
where \(\Delta s_{iz}\) is the \"local shear displacement\" caused by the interface of pile element \(i\) in a pile group. \(W_{isz}\) is the elastic soil displacement outside the disturbed soil around pile element \(i\) in the pile group. [PERSON] and [PERSON] [20] proposed to represent the \"local shear displacement\", \(\Delta s_{iz}\), with a hyperbolic nonlinear relationship in the following form:  
\[\tau_{iz}=\frac{\Delta s_{iz}}{a+b\Delta s_{iz}}, \tag{2}\]  
in which \(\tau_{iz}\) is the shaft shear stress caused by a pile element \(i\) shaft interface, \(a\) and \(b\) describe the empirical factors of the soil interface. \(a\) and \(b\) can be determined by experimental or back-analysis methods. The physical meaning of coefficient, \(a\), represents the reciprocal of the initial stiffness, \(K_{si}\), which is the relationship of the pile-soil interface that is caused by shear stress and local shear displacement (see Figure 2). The units of the force per cubic length can also be represented as follows:  
\[K_{si}=\frac{1}{a}, \tag{3}\]  
The inverse of the coefficient \(b\) is the asymptotic shear stress, demonstrated by \(\tau_{ult}\) (i.e., \(\tau_{ult}=1/b\)), which is obtained by the shear stress-displacement curve with a very large local shear displacement value (see Figure 2). The failure strength of a pile-soil interface, \(\tau_{f}\), is slightly lower than the asymptomatic shear stress, \(\tau_{ult}\). By introducing a failure ratio, \(R_{f}\), to describe it, the calculation can be performed using the following formula:  
\[\tau_{f}=R_{f}\tau_{ult}, \tag{4}\]  
According to [PERSON] and [PERSON]'s [22] research, the relative displacement curve of the shear stress is shown in Figure 2, where the \(R_{f}\) value is between 0.8 and 0.95. Figure 1: (**a**) Idealised displacements developed around the pile element; (**b**) A simplified approximation to model the disturbed soil zone. According to coefficients \(a\) and \(b\) given above, two parameters \(K_{si}\) and \(\tau_{f}\) need to properly define the behaviours caused by the pile-soil interface. Both the initial shear stiffness, \(K_{si}\), and the shear strength of the interface, \(\tau_{f}\), are dependent on the normal stress value of the interaction on the interface.