# Effect of Characteristics of Ground Motion on Seismically Induced Sliding Surface of Slopes
## 2 Material and Methods  
2D dynamic nonlinear analyses were performed using the commercial FE program LS-DYNA (2D) [PERSON] et al. (2018) to simulate the non-linear response of slope, as shown in Figure 1. The program uses the explicit central difference time integration scheme in solving the dynamic equation of motion. The semi-discrete equation of motion at time n is shown in Equation (1). Where diagonal mass matrix is represented by \(M\), external and body force load is accounted as \(P^{n}\), stress divergence vector as \(F^{n}\) and hourglass resistance as \(H^{n}\)[19]. \[Ma^{n}=P^{n}-F^{n}+H^{n} \tag{1}\]  
The dimensions of the numerical model were 10 m slope height, 27 m wide, and 34.65 m in depth. For simulation of the rigid base, a prescribed motion boundary condition was assigned to the base nodes of the slopes. Due to the difference in the height of vertical edges of the soil domain, the equal degree of freedom (EDOF) constraint was not applied. Instead, the lateral boundaries were fixed in the horizontal direction. A sensitivity study was performed to select the extent of lateral boundaries, such that they do not influence the response of the slopes due to the reflected waves. Conservatively, the distance from the slope to both ends of the lateral boundaries was set to 105 m.  
The soil domain was discretized using four noded-plane strain elements. The size of the smallest element was selected such that the maximum wave frequency (\(f_{max}\)) is higher than 25 Hz as recommended for the site response analysis [20; 21]. \(f_{max}\) was calculated as \(f_{max}=Vs/4\)h, where h is the height of the element and \(Vs\) is the shear wave velocity. The element size selected was 0.5 m, which is also smaller than recommended by [PERSON] and [PERSON] [22]. The slope area was discretized using a fine mesh. The mesh size gradually increases towards the lateral boundary to reduce the computational cost without affecting the numerical accuracy. In the computational model, slope was discretized into layers such that, within each layer, the confining pressure is within a narrow range. Therefore, inclined layers were used for the slope, as shown in Figure 1. This type of discretization was necessary because \(Vs\) is influenced by the confining pressure. The nested-yield surface plasticity model implemented in LS DYNA was used to simulate the nonlinear behavior of soil (MAT_HYSTERETIC model). The non-linear shear stiffness is represented in a piecewise manner by separating the shear behavior across multiple linear elastic perfectly plastic yield surfaces. The elastic stiffness and size of the yield surfaces are controlled by a backbone curve of the shear stress versus shear strain at a reference pressure. Modulus degradation and Masing rule damping are determined internally based on the user-defined stress-strain inputs. The value of stress at any instant is equal to the sum of the stresses in the individual elastic-perfect plastic layers, which results in a piecewise backbone curve. It has been widely used for seismic site response analyses [21; 23; 24]. The properties are pressure dependent on descrambling by the following equations [19]:  
\[G(p)=\frac{G_{0}(p-p_{0})^{b}}{\left(p_{ref}-p_{0}\right)^{b}} \tag{2}\]  
\[\frac{\tau(p)}{\tau\left(p_{ref}\right)}=\sqrt{\frac{a_{0}+a_{1}(p-p_{0})+a_{2 }(p-p_{0})^{2}}{a_{0}+a_{1}\left(p_{ref}-p_{0}\right)+a_{2}(p_{ref}-p_{0})^{2}}} \tag{3}\]  
where \(p\) is the current pressure, \(p_{o}\) is the cut-off or datum pressure, \(p_{ref}\) is the reference pressure at which input value are calculated, \(G_{o}\) is the shear modulus of the layer at the reference pressure, \(G(p)\) is the shear modulus of the layer at pressure \(p\), \(\tau(p_{ref})\) is the  
Figure 1: Numerical model of the slope. yield stress of the layer at the reference pressure, \(\tau(p)\) is the yield stress of the layers at pressure \(p\), \(b\) is the exponent for pressure sensitive moduli and \(a_{0}\), \(a_{1}\) and \(a_{2}\) are the yield function constants.