# Dynamic Behavior of an Embedded Foundation under Horizontal Vibration in a Poroelastic Half-Space
## 4 Conclusions  
A simplified analytical method was performed to investigate the dynamic response of embedded foundation under horizontal vibration in the poroelastic soil medium. [PERSON]'s theory was used to solve the dynamic behavior of soil. The mixed boundary value problem was analyzed by adopting a simplified analytical method. The proposed method in this study was confirmed by comparing the results with the existing literature. Analytical expressions were developed for predicting the dynamic impedance of embedded foundation under horizontal vibration. Furthermore, a comprehensive parametric analysis was performed in order to assess the applicability of this method. The dynamic impedance and dynamic response factor versus frequency were depicted in the dimensionless scale, respectively. Some conclusions are drawn below:  
1. Results illustrated in this paper show that the embedment ratio of the foundation, mass ratio of the foundation, relative stiffness, and poroelastic properties of the backfill has a significant influence on the horizontal dynamic response of embedded foundation for the frequency range of \(0\leq a_{0}\leq 5\).
2. The horizontal dynamic impedance of embedded foundation increases while the dynamic response factor decreases with the increasing embedment ratio. There's an increase in dynamic impedance and a decrease in dynamic response factor with the increasing frequency.
3. It should be noted that the real part of dynamic impedance increases with the increasing mass ratio while the imaginary part is independent of the mass ratio. The dynamic response factor decreases with increasing mass ratio. An increase in the dynamic impedance and a decrease in the dynamic response factor are described with the increasing frequency.  
Figure 11: Dynamic response factor of embedded foundation under different soil stiffness.  
By increasing the poroelastic material properties, the real part of dynamic impedance increases while the imaginary part, as well as the dynamic response factor, decreases. The larger the frequency, the larger is the dynamic impedance and the smaller the dynamic response factor.
5. The dynamic impedance increases significantly, especially for a frequency smaller than 1.5. The real part keeps increasing while the imaginary part decreases when the frequency is larger than 1.5. A considerable decrease in the dynamic response factor is noted due to the presence of relative backfill modulus. The effect of [PERSON]'s ratio \(v\) on the horizontal dynamic impedance of embedded foundation can be ignored.  
With the aid of simplified analytical solutions proposed in this study, the expressions of dynamic impedance and dynamic response factor are formulated. The method proposed in this paper can be used to evaluate the dynamic response of embedded foundation subjected to horizontal dynamic loading. The comprehensive parametric study can provide versatile guidance to civil engineering and other dynamic construction. However, the proposed method is limited by the perfect smooth contact between the foundation and soil interface. The analytical solution of the horizontal impedance of embedded foundation with rigid underlying bedrock also remains unclear.  
Concepts and designs: Y.C. and W.Z.; Data analysis and numerical results: P.J. and J.H.; Examination of the simulation results: Y.G.  
The authors would like to thank Northeastern University in providing access to the software and other facilities. The research described in this paper was supported by the National Science Foundation of China (No. 51578116, No. 51878127) and the China Scholarship Council (No. 201706080075). These supports are gratefully acknowledged. And the data is available for the journal.  
The authors declare no conflict of interest.