## 2 Methods
### The Back-Analysis Method Procedure and Analysis Results  
In theory, laboratory tests, such as ring shear tests, can be used to obtain \(a\) and \(b\), which are the parameters of the recommended hyperbolic model. Nevertheless, it is difficult for laboratory tests to restore the relative displacement characteristics of the actual pile-soil interface in the field. At the same time, the relative displacement characteristics of the materials and soil contact interfaces are not obvious and have significant effects on \(a\) and \(b\), such as the roughness of the pile-soil interface, the piling method, and the soil type. Therefore, the use of back-analysis methods can effectively improve the accuracy and rationality of the a and b values, and the back-analysis approach is given as follows:  
The depth, \(p(z)\), and load distribution results are obtained from the pile load test data measured in the field. The interpolated spline curve is used to fit the measurement data to obtain a load distribution with a depth function, \(\overline{p}(z)\);  
According to the spline fitting function, \(\overline{p}(z)\), the shaft shear stresses are calculated by the following formula:  
\[\tau_{z}=-\frac{1}{2\pi\tau_{o}}\frac{d\overline{p}(z)}{dz}, \tag{12}\]  
Equation (12) can also be used to determine the failure strength of a pile-soil interface \(\tau_{f}(z)\) at depth \(z\). The value of \(b\) at various depths, \(z\), can also be calculated as follows:  
\[b(z)=R_{f}/\tau_{f}(z), \tag{13}\]The shaft displacement, \(s_{z}\), at depth, \(z\), can also be calculated as follows:  
\[s_{z}=s_{ti}-\frac{1}{E_{p}A_{p}}\int\limits_{0}^{z}\overline{p}(z)dz, \tag{14}\]  
where \(E_{p}\) is the elastic modulus of the pile, \(A_{p}\) is the cross-sectional area of the pile, and \(s_{ti}\) is the displacement of the pile head at the different load steps. According to Equation (14), the value of \(a\) can be calculated as follows:  
\[a(z)=\frac{(s_{z}-c\tau_{z})(1-b(z)\tau_{z})}{\tau_{z}}, \tag{15}\]  
Based on Equations (13) and (15), \(a(z)\) and \(b(z)\) can be determined. Comparison of the Simplified Approach with Boundary Integral Solutions and the Elastic Approximate Approach  
The details of the algorithm and programming procedure are described in the work by [PERSON] and [PERSON] [21]. The flowchart of the proposed method is presented in Figure 3. To verify that the simplified nonlinear approach and the algorithm proposed by [PERSON] and [PERSON] [21] are useful in the analysis of pile groups, in this paper, comparisons are conducted with the results predicted by the more stringent boundary integral solution proposed by [PERSON] and [PERSON] [5] and by the elastic approximate approach projections of [PERSON] and [PERSON] [7]. In the comparison, assuming that the cap is rigid and not in direct contact with the ground, the stiffness ratio, \(\lambda=(E_{p}/G_{s})\), is equal to 1000. Since the other two methods only predict the elastic behaviour of the pile groups, the total load applied on a rigid cap is only 10% of the ultimate total load in this study. This will ensure that the pile group behaviour predicted by the proposed analysis is largely in the elastic stage. Figure 4 graphically presents a comparison of the results in terms of the normalised stiffness (\(P_{i}/\omega r_{o}G_{s}\)) versus the pile group slenderness ratio (\(L/D\)), where \(P_{i}\) is the sensing vertical load on the single pile head of the 3 \(\times\) 3 pile groups, \(w\) is the displacement of  
Figure 3: Flowchart for study load-settlement behaviour of pile group. a pile group, and \(D\) and \(L\) are the pile diameter and the length of the pile, respectively. The normalised spacing (\(S\)/\(D\)) of the pile groups is equal to 3. It can be observed that very good agreements are obtained between the simplified method, the boundary integral solution, and the elastic approximate approach.