# Dynamic Behavior of an Embedded Foundation under Horizontal Vibration in a Poroelastic Half-Space
## 3 Results and Discussion
### Effect of the Footing Mass Ratio  
The mass ratio can significantly affect the dynamic response of embedded foundation. Figure 6 shows the dynamic response of embedded foundation in order to investigate the effect of the mass ratio of the foundation under horizontal vibration with a depth ratio \(L/R=1.0\). In general, the dynamic impedance of the embedded foundation increases by increasing the mass ratio. The initial density of embedded foundation is set to be one, and the density of massive foundation varies from one to four in order to examine the mass ratio of the embedded foundation. It is obvious that an increase in the mass ratio can definitely increase the stiffness of foundation, as shown in Figure 6. The foundation  
Figure 4: Horizontal dynamic impedance of embedded foundation under different embedment ratio.  
Figure 5: Dynamic response factor of embedded foundation under different embedment ratio.  
stiffness increases almost linearly with the increasing value of \(a_{0}\) when the mass ratio (\(RM\)) is 1:4, with almost the same speed. The increasing trend tends to be gentle when \(a_{0}>1.5\). The mass ratio has little effect on the imaginary part of \(K_{\rm h}\). The reason for this phenomenon is that the soil mass ratio has no influence on the contact with the surrounding soil. The imaginary part increases first to a maximum value at about \(a_{0}=1.5\) and then decreases for a larger value of frequency. The imaginary part of horizontal impedance \(K_{\rm h}\) is nearly independent of the mass ratio when compared to the real part. The imaginary part shows a gradual increase when \(a_{0}<1.5\), which shows a near linear variation with \(a_{0}\). The mass ratio is more significant on the real part of the horizontal impedance and relatively smaller on the imaginary part of \(K_{\rm h}\). The parametric study indicates that the imaginary part of horizontal impedance is less affected by the mass ratio than the real part.  
Figure 7 shows the influence of mass ratio on the horizontal dynamic response factor \(A_{\rm h}\) for an embedment ratio \(L/R=1\). It can be concluded that the mass ratio of the embedded foundation has a significant influence on the dynamic response factor. The peak value of the dynamic response factor is at zero frequency, and the foundation stiffness is the maximum at this state. By increasing the value of the mass ratio, the dynamic response factor decreases, and the foundation stiffness increases. In addition, the dynamic response factor decreases sharply with the lower value of the mass ratio. The dynamic response of foundation also decreases with the increase in nondimensional frequency \(a_{0}\). The value of \(A_{\rm h}\) is close to each other at any mass ratio when the frequency is larger than 3. It is noted that a significant decrease occurs when \(a_{0}<1.5\), and then the dynamic response factor decreases slowly as \(a_{0}\) increases. As expected, the response factor decreases with increasing foundation mass. However, at high frequencies away from 1.5, the response factor shows little dependence on the mass ratio.  
Figure 6: Horizontal dynamic impedance of embedded foundation under different mass ratio.  
Figure 7: Dynamic response factor of embedded foundation under different mass ratio.