# Numerical Algorithm for Dynamic Impedance of Bridge Pile-Group Foundation and Its Validation
## 2 Project-Oriented Numerical Algorithm for Impedance of Pile-Group Foundation  
### Definition of the Impedance Functions  
For a generic bridge founded on pile groups, the soil-structure interaction problem can be conveniently studied in the frequency domain by assuming a linear behavior for soil and structure. By adopting a finite element method, the following system of complex linear equations governs the dynamics of the whole system [3]:  
\[\mathbf{Z}(\omega)\mathbf{d}(\omega)=\mathbf{f}(\omega) \tag{1}\]  
where \(\mathbf{Z}\) is the dynamic stiffness matrix of the soil-foundation-superstructure system; \(\mathbf{d}\) is the vector of the nodal displacements; and \(\mathbf{f}\) is the vector of nodal forces. With the assumption that the caps connecting the pile heads are rigid, and assigning different master nodes to them, Equation (1) can be partitioned as:  
\[\left[ \begin{array}{ccc}\mathbf{Z}_{\text{ss}}&\mathbf{Z}_{\text{sf}}&0\\ \mathbf{Z}_{\text{fs}}&\mathbf{Z}_{\text{tf,s}}+\mathbf{Z}_{\text{ff,f}}&\mathbf{Z}_{\text{fe} }\\ 0&\mathbf{Z}_{\text{ef}}&\mathbf{Z}_{\text{ee}}\end{array} \right]\left[ \begin{array}{ c}\mathbf{d}_{\text{s}}\\ \mathbf{d}_{\text{f}}\\ \mathbf{d}_{\text{e}}\end{array} \right]=\left[ \begin{array}{c}0\\ \mathbf{f}_{\text{f}}\\ \mathbf{f}_{\text{e}}\end{array} \right] \tag{2}\]  
in which the subscripts s, f, and e denote the superstructure, the rigid pile cap, and the piles, respectively. It is worth noting that no displacement component for the soil appears in Equation (2) because the pile-soil interaction is accounted for in the definition of the dynamic stiffness matrices relevant to the rigid caps and piles [3]. With the linear assumption, and according to the substructure approach, the inertial interaction for the superstructure is governed by:  
\[\left[ \begin{array}{ccc}\mathbf{Z}_{\text{ss}}&\mathbf{Z}_{\text{sf}}\\ \mathbf{Z}_{\text{fs}}&\mathbf{Z}_{\text{ff,s}}+\mathbf{S}\end{array} \right]\left[ \begin{array} []{c}\mathbf{d}_{\text{s}}\\ \mathbf{d}_{\text{f}}\end{array} \right]=\left[ \begin{array}{c}0\\ \mathbf{S}\mathbf{d}_{\text{FIM}}\end{array} \right] \tag{3}\]  
and for the soil-foundation system, the kinematic interaction problem is governed by:  
\[\left[ \begin{array}{ccc}\mathbf{Z}_{\text{ff,f}}&\mathbf{Z}_{\text{fe}}\\ \mathbf{Z}_{\text{ef}}&\mathbf{Z}_{\text{ee}}\end{array} \right]\left[ \begin{array}{ c}\mathbf{d}_{\text{FIM}}\\ \mathbf{d}_{\text{e,kin}}\end{array} \right]=\left[ \begin{array}{c}\mathbf{f}_{\text{ F}}\\ \mathbf{f}_{\text{E}}\end{array} \right] \tag{4}\]  
where \(\mathbf{d}_{\text{e,kin}}\) are the displacements of the embedded piles and \(\mathbf{d}_{\text{FIM}}\) are those of the master modes of the rigid caps due to the kinematic interaction. Then, the impedance matrix of the soil-structure system \(\mathbf{S}\) can be expressed as:  
\[\mathbf{S}(\omega)=\mathbf{Z}_{\text{ff,f}}-\mathbf{Z}_{\text{te}}\mathbf{Z}_{\text{ee}}^{-1} \mathbf{Z}_{\text{ef}} \tag{5}\]  
It can be found that foundation impedance functions can be defined as the ratio of a harmonic force (or a moment) applied on the foundation-soil interface to the corresponding harmonic displacement (or rotation), dependent not only on the frequency of the excitation, but also on the foundation geometry and the dynamic characteristics of the soil medium [12]. Based on this definition, a project-oriented numerical algorithm for impedance functionsof bridge pile-group foundation is proposed.