## 5 Method of Determining Parameters by Field Experiments
### Parameter Identification of the Circular Flexible Bearing Plate Test  
Jinping I Hydropower Station uses flexible bearing plates with a diameter of 105 cm (effective output size of 100 cm) at the dam site to conduct creep tests on the bearing plates on site [8]. A displacement measuring instrument was installed at the edge of a circular flexible bearing plate, and the elastic and viscoelastic solutions of settlement at the measuring point were calculated using Equations (8) and (12), in which F\((x,y)\) is expressed in polar coordinates as follows:  
\[F(x,y)=F(r=R_{0})=4 pR_{0} \tag{19}\]  
where \(R_{0}\) is the radius of the pressure plate, \(r\) is the pole diameter, and \(p\) is the uniform load under the bearing plate. Figure 8 shows the creep curves of the rock mass at different loading stages. The figure shows that the rock mass has obvious instantaneous abrupt displacement and tends to be stable after creep with a gradually decreasing rate. Therefore, the rock mass can be regarded as a viscoelastic body, and its deformation can be analysed using the fractional-order generalised Kelvin model, including instantaneous and creep deformations. The process of parameter inversion is shown in Figure 9. First, based on the elastic displacement formula at the edge of a circular flexible bearing plate, the elastic modulus \(E_{0}\) of the rock mass can be obtained from the instantaneous deformation, \(w_{0}\), at the edge of the bearing plate:  
\[E_{0}=\frac{4 pR_{0}(1-\mu^{2})}{\pi w_{0}} \tag{20}\]  
Figure 8: Creep curves of rock mass at different loading stages of circular flexible pressure-bearing plates. The shear and bulk moduli of the rock mass are then expressed as follows:  
\[G_{1}=\frac{E_{0}}{2(1+\mu)},K=\frac{E_{0}}{3(1-2\mu)} \tag{21}\]  
Second, other parameters, \(G_{2}\), \(\eta\) and \(\alpha\), in the fractional-order model are inverted according to the measured creep displacement. The objective function is established by the residuals of the measured displacement, \(w(t_{i})\), and the calculated displacement, \(w(X,t_{i})\), at moment \(t_{i}\), and the objective function in the form of the sum of squares is expressed in this study as follows:  
\[F(X)=\sum_{i=1}^{N}\left[w(t_{i})-w(X,t_{i})\right]^{2} \tag{22}\]  
Figure 9: The process of parameter inversion. where \(X\) is the design variable expressed as follows:  
\[X=(X_{1},X_{2},X_{3})=(G_{2},\eta,\alpha) \tag{23}\]  
where \(N\) is the number of measured displacements. In this study, the set of \(X^{*}\) that minimises Equation (22) is obtained using the differential evolution method [32], and \(X^{*}\) is the determined parameter. This degenerates to the case of using the integer-order generalised Kelvin model if \(\alpha=1\), and the parameters \(G_{2}\) and \(\eta\) can be obtained using the differential evolution method. In this study, the coefficient of determination, \(R^{2}\), is applied to evaluate the performance of the aforementioned two models. \[R^{2}=1-\frac{\sum\limits_{i=1}^{N}[w(t_{i})-w(X^{*},t_{i})]^{2}}{\sum\limits _{i=1}^{N}[w(t_{i})-\overline{w}]^{2}} \tag{24}\]  
where \(\overline{w}\) is the average value of the measured displacements. Tables 2 and 3 show the parameters of the rheological models obtained under different loading conditions, based on the above two-step inversion analysis process. The theoretical displacement curves under different stress levels are then obtained by substituting the inversion analysis parameters into the corresponding analytical equations. Figure 10 shows the comparisons between the theoretical displacement curves based on the two rheological models and the measurements obtained under different stress levels.