## 3 Dynamic Response Analysis Method
### Computational Model  
In static analyses, fixed or elastic boundaries can be realistically placed at some distance from the region of interest. In dynamic problems, however, such boundary conditions cause the reflection of outward propagating waves back into the model and do not allow the necessary energy radiation. The use of a larger model can minimize the problem, since material damping will absorb most of the energy in the waves reflected from distant boundaries. However, this solution leads to a large computational burden [22]. At the same time, the slope simulated here is not a structure, but a part of the geological body, and how large a model is needed to accurately model the behavior of the slope is not clear yet [23]. Thus, the alternative is to use quiet (or absorbing) boundaries. The viscous boundaries developed by [PERSON] and [PERSON] [24] are used in our simulations; they are placed at the left side, right side, and bottom of the model in the normal and shear directions (Figure 7). The dashpots provide viscous normal and shear tractions [22]:  
\[t_{n}=-\rho C_{p}v_{n} \tag{7}\]  
\[t_{s}=-\rho C_{S}v_{s} \tag{8}\]  
where \(v_{n}\) and \(v_{s}\) are the normal and shear components of the velocity at the boundary, \(\rho\) is the mass density, and \(C_{P}\) and \(C_{S}\) are the P and Sv wave velocities.  
\(C_{P}\) is given by:  
\[C_{P}=[E(1-v)/(\rho(1+v)(1-2v))]^{0.5} \tag{9}\]and \(C_{S}\) is given by:  
\[C_{S}=[E/2\rho(1+\
u)]^{0.5}=(G/\rho)^{0.5} \tag{10}\]  
where \(E\) is the dynamic elastic modulus, \(G\) is the dynamic shear modulus, and \(\
u\) is the Poisson's ratio.  
The slope is meshed into brick elements [22]. The size of elements has a great influence on the computational results. [PERSON] and [PERSON] [24] show that, for accurate representation of wave transmission through a model, the spatial element size must be smaller than approximately one-tenth to one-eighth of the wavelength associated with the highest frequency component of the input wave. In this paper, to balance between accuracy and efficiency, the ratio of the mesh size to the wavelength is equal to \(1/12\). In addition, the deformational behavior of joints is assumed to be linearly elastic both in the normal and shear directions, and the rock material is assumed to be elastic, isotropic, and homogeneous.  
To the viscous boundary, the dynamic input must be applied in a stress or force form, or the effect of the viscous boundary would be nullified. To input seismic motion at a quiet boundary, a stress boundary condition is used (i.e., a velocity or acceleration record is transformed into a stress record and applied to a quiet boundary). A velocity wave may be converted to an applied stress using the formula:  
\[\sigma_{n}=2(\rho C_{P})\;v_{n} \tag{11}\]  
or  
\[\sigma_{s}=2(\rho C_{s})\;v_{s} \tag{12}\]  
where \(\sigma_{n}\) is the applied normal stress and \(\sigma_{S}\) is the applied shear stress; the factor of two in Equations (11) and (12) accounts for the fact that the applied stress must be double that observed in an infinite medium, since half the input energy is absorbed by the viscous boundary.  
In this study, one shear wave \(a=0.5g\mathrm{cos}(2\pi t/T)\) is applied under the bottom of the slope. According to the character described above, the acceleration time history is converted to a stress-time history:  
\[\sigma_{s}=(gT/2\pi)(E\rho/(2(1+\
u)))^{0.5}\mathrm{sin}(2\pi t/T) \tag{13}\]  
where \(T\) is the period of the wave and \(g\) is the gravity acceleration.