###### Contents  
* 1 Introduction
\t* 2.1 Background of traffic infrastructures
\t* 2.2 The role of the traffic infrastructure
\t* 2.3 The role of the traffic infrastructure
\t* 2.4 The role of the traffic infrastructure
\t* 2.5 The role of the traffic infrastructure
\t* 2.6 The role of the traffic infrastructure
\t* 2.7 The role of the traffic infrastructure
\t* 2.8 The role of the traffic infrastructure
\t* 2.9 The role of the traffic infrastructure
\t* 2.1 The role of the traffic infrastructure  
SLT in the laboratory using six model piles. In addition to the SLT, dynamic testing can also be employed to estimate the pile capacity. The high-strain dynamic pile testing (HSDPT) method is generally used to evaluate the capacity of a deep foundation [26; 27]. However, both SLT and HSDPT are difficult to use on existing bridge foundations that are in service, as permanent displacements can be generated by these methods. For bridges with potential diseases or for bearing increasing train loads and train speeds, it is considered significant that engineers learn the real foundation capacities to determine whether the foundations need to be strengthened. Then, the transient response method (TRM) and index of dynamic stiffness (\(K_{\text{d}}\)) are proven to be feasible. The TRM is also known as the mechanical mobility method. By applying a vertical impact load on the foundation, both the velocity responses and force signals can be analysed in the frequency domain. The mechanical admittance spectrum or mobility can be obtained using the velocity and force spectra [28]. Using the TRM, the dynamic stiffness can be obtained as follows [29; 30; 31]:  
\[K_{\text{d}}(f)=\frac{2\pi f}{|V(f)/F(f)|}, \tag{1}\]  
where \(V(f)\) is the Fourier spectrum of the velocity response \(v(t)\), \(F(f)\) is the Fourier spectrum of the force \(F(t)\), and the ratio of \(V(f)/F(f)\) is the mechanical admittance spectrum or velocity mobility [28]. Figure 1 shows the theoretical velocity mobility of a pile obtained using the TRM. In the low-frequency range, the ideal velocity mobility varies linearly with the frequency; accordingly, the value of \(K_{\text{d}}\) in the low-frequency range is a constant. Recently, the TRM and \(K_{\text{d}}\) were employed for pile foundation evaluations. [PERSON] and [PERSON] [32] proved that in the pile capacity estimation, a reasonable value of \(K_{\text{d}}\) could be employed as an alert value. [PERSON] et al. [33] indicated that \(K_{\text{d}}\) could be employed to estimate the pile bearing capacity. [PERSON] et al. [34] investigated the influence of the cap-pile system on \(K_{\text{d}}\). [PERSON] and [PERSON] [35] performed both TRM and SLT on a full-scale pier-cap-pile system and analysed the relationship between the dynamic stiffness and foundation capacity. However, the above studies only analysed the applicability of the TRM to pile foundations; other types of bridge foundations, such as the widely used spread foundation and calsson foundation, were not considered. In addition, most studies have only considered the TRM; a few tests were reported on the use of both TRM and SLT on the same foundation because of low feasibility of the SLT to evaluate in-service pier foundations and the associated high testing cost. If the TRM and \(K_{\text{d}}\) are used to evaluate the capacities of different types of bridge foundations, then the relationships between the dynamic stiffnesses and bearing capacities could be further investigated. Therefore, in this study, a model test was performed in the laboratory, where both the TRM and SLT were used to assess three types of bridge foundations: spread, calsson, and grouped pile foundations. The dynamic stiffnesses were analysed when the foundations were first fully buried in the soils and then when the constraint states were changed. Subsequently, the static stiffnesses were calculated  
Figure 1: Theoretical mobility response spectrum tested using the TRM. All the models were cast in situ using self-mixed concrete.