# Dynamic Behavior of an Embedded Foundation under Horizontal Vibration in a Poroelastic Half-Space
## 3 Results and Discussion
### Effect of the Relative Stiffness  
The effect of relative stiffness on the horizontal dynamic impedance of embedded footing in poroelastic soil medium are investigated to demonstrate the influence of nondimensional frequency. Figure 10 shows that the dynamic impedance \(K_{\mathrm{h}}\) varies with frequency \(a_{0}\) with a different value of the relative soil stiffness. It is noted that the horizontal dynamic impedance depends on the relative soil stiffness with other fixed parameters. The real part of \(K_{\mathrm{h}}\) increases with the increasing \(G_{\mathrm{s}}/G_{\mathrm{b}}\) which is due to the embedded foundation. The foundation stiffness enhances with larger soil stiffness around the foundation. The foundation stiffness also increases with the increasing frequency, especially for a frequency higher than 1. In general, the imaginary part of dynamic impedance increases with the increasing \(G_{\mathrm{s}}/G_{\mathrm{b}}\), while this trend reverses when \(a_{0}>4\). The damping of foundation increases to a maximum value at about \(a_{0}=2\) and then decreases at the same speed for different \(G_{\mathrm{s}}/G_{\mathrm{b}}\) with the increasing \(a_{0}\). Figure 10 also indicates that the relative stiffness of the foundation soil has a noticeable influence on the value of dynamic impedance. Both foundation stiffness and damping increase as the \(G_{\mathrm{s}}/G_{\mathrm{b}}\) increases. The results from Figure 10 also show that the relative stiffness of the subsoil has a strong effect on the magnitude of the horizontal impedance \(K_{\mathrm{h}}\), which is investigated by discussing the real and imaginary part separately. Another important conclusion revealed from Figure 10 is the rising difference of the dynamic impedance between the undisturbed soil and disturbed soil which has also been proved by [PERSON] et al [20].  
Figure 10: Horizontal dynamic impedance of embedded foundation under different soil stiffness.  
Figure 9: Dynamic response factor of embedded foundation under different poroelastic material properties.  
The dynamic response factor of embedded foundation under horizontal excitation under different relative stiffness is investigated by varying \(a_{0}\). It is indicated that the response factor decreases with the increasing \(G_{\mathrm{s}}/G_{\mathrm{b}}\), especially when \(G_{\mathrm{s}}/G_{\mathrm{b}}\) increases from 0.1 to 0.5. As expected, the foundation stiffness increases with the increasing \(G_{\mathrm{s}}/G_{\mathrm{b}}\) because of the side-wall effect of embedded foundations. The dynamic response factor decreases with the increasing frequency at the range of 0 to five. Note that \(A_{\mathrm{h}}\) decreases significantly at a lower frequency, \(a_{0}<1.5\), while the dynamic response factor shows little dependence on \(a_{0}\) when \(a_{0}>1.5\). It is evident that the horizontal dynamic response factor depends significantly on the frequency and the relative stiffness of subsoil and backfill. The dynamic response factor shows a similar trend for different values of \(G_{\mathrm{s}}/G_{\mathrm{b}}\). Another important conclusion revealed in Figure 11 is the independence of the dynamic response factor on high frequency.