# Numerical Algorithm for Dynamic Impedance of Bridge Pile-Group Foundation and Its Validation
## 3 Verification of the Numerical Algorithm
### Comparison with Existing Methods  
First, a 4 \(\times\) 4 pile-group foundation was chosen as an example, the parameter values of which are taken from [15] and shown in Table 1. For comparison, both the thin layer method presented in [15] and the proposed algorithm were used to obtain the impedance functions of the single pile as well as the pile-group foundation. The overall size of the FE model in the proposed algorithm is 80 m \(\times\) 80 m \(\times\) 80 m, and the size of the soil element is set as 2.5 m. The damping of the piles is neglected in accordance with the benchmark. Figures 3 and 4 show the calculated vertical impedance functions of the single pile and the pile-group foundation, respectively, as obtained by the proposed algorithm and the thin layer method. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Item** & **Parameter** & **Value** \\ \hline \multirow{5}{*}{Piles} & Pile diameter & 1.0 m \\  & Pile length & 20 m \\  & Pile spacing & 5.0 m \\  & Unit weight & 20 kN/m\({}^{3}\) \\  & Poisson’s ratio & 0.25 \\  & Elasticity modulus & \(3.92\times 10^{9}\) N/m\({}^{2}\) \\ \hline \multirow{5}{*}{Soil} & Unit weight & 14 kN/m\({}^{3}\) \\  & Damping ratio & 0.05 \\ \cline{1-1}  & Velocity of shear wave & 100 m/s \\ \cline{1-1}  & Poisson’s ratio & 0.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the 4 \(\times\) 4 pile-group foundation. The impedance functions are expressed as \(k+ia_{0}c\), where \(k\) and \(c\) are the stiffness item and the damping item, respectively. \(k_{\mathrm{VV}}^{S}\) represents the vertical static stiffness of a single pile in elastic half-space, while \(a_{0}\) is the dimensionless frequency, which can be expressed as:  
\[a_{0}=\frac{\omega d}{V_{\mathrm{s}}} \tag{14}\]  
where \(\omega\) is the frequency; and \(d\) and \(V_{\mathrm{s}}\) are the pile diameter and shear wave velocity of the soil, respectively. It can be seen that the results obtained by the proposed algorithm are consistent with those obtained by the thin layer method. Even though there is some error, which probably occurs due to the relatively large element size or the difference between calculation assumptions, from the perspective of practical application, the results are sufficiently accurate to meet the requirements of engineering. Figure 4: Vertical impedance function of a pile-group foundation compared with the thin layer method. Figure 3: Vertical impedance function of a single pile compared with the thin layer method. Second, the results obtained by the proposed algorithm were compared with those obtained through a rigorous method presented in a recent study [39] based on a \(4\times 4\) pile-group foundation. For the results presented in this work, it was assumed that \(v_{\mathrm{s}}\) ([PERSON]'s ratio of the soil) = 0.4, \(E_{\mathrm{pile}}/E_{\mathrm{soil}}\) = 100 (\(E_{\mathrm{pile}}\) and \(E_{\mathrm{soil}}\) are the elasticity moduli of the piles and soil, respectively), \(v_{\mathrm{p}}\) ([PERSON]'s ratio of the piles) = 0.25, \(L\) (length of the piles) = 15\(d\) (where \(d\) is the pile diameter), \(S\) (pile spacing) = 5\(d\), and \(\rho_{\mathrm{s}}/\rho_{\mathrm{p}}\) = 0.7, where \(\rho_{\mathrm{s}}\) and \(\rho_{\mathrm{p}}\) are the mass density of the soil and the piles, respectively. Figure 5 shows the calculated vertical impedance functions of the pile-group foundation, as obtained by the proposed algorithm and the existing rigorous method. It should be noted that the symbols in Figure 5 have the same meanings as those in Figure 4. It can be seen that the results obtained by the proposed algorithm are also consistent with those obtained by the rigorous method. The error of the maximum stiffness item is 0.6% and 4% as calculated from Figures 4 and 5, respectively, falling within an acceptable range of values for engineering projects.