# Estimation of Seismically-Induced Crest Settlement of Earth Core Rockfill Dams
## 2 Numerical Model  
A series of 2D dynamic analyses was performed using the finite difference analysis program FLAC2D (Fast Lagrangian Analysis of Continua in two dimensions), Version 7.0 [5] to calculate the settlement at the crest of an ECRD. The baseline numerical model of the ECRD is shown in Figure 1. The dam was 60 m and 260 m in height and width, respectively. The dimensions of the dam were assigned based on actual ECRDs constructed in Korea. The upstream and downstream slopes were set to 1:2.2 and 1:2.0, respectively. The slope of the core was set to 1:0.2. The thickness and width of the soil model below the dam were set to 40 m and 400 m, respectively. The shear wave velocity profiles of the rockfill and core sections were constructed from the empirical shear wave velocity equations of [PERSON] and Takahashi [6], as shown in Figure 2. The core profile lay between the lower and upper bounds of [PERSON] and [PERSON]'s [6] equations for the core. The rockfill profile also fell between the saturated and unsaturated rockfill profiles of Sawada and Takahashi [6]. The applicability of the empirical equations of [PERSON] and [PERSON] [6] was confirmed using geophysical measurements [7] and inversion analysis results derived from artificial blast recordings [8]. Table 1 lists the properties used for the rockfill and the core. For the underlying foundation soil, the shear wave velocity was set at 1100 m/s.  
The nonlinearity of the soil was reported to play an important role in the seismic response of slopes [9]. The Sig3 model, which is one of the most widely used nonlinear soil models among those included in the library of FLAC\({}^{\text{2D}}\), was used to simulate the nonlinear behaviors of the rockfill and the core. The Sig3 model was defined as follows:  
\[M_{s}=\frac{a}{1+\exp(-(L-x_{0})/b)} \tag{1}\]  
where \(M_{s}\) is the secant shear modulus, \(L\) is \(\log(\gamma)\), and \(x_{o}\), \(a\), and \(b\) are curve-fitting parameters. Sig3 was used in a number of studies to represent the nonlinear soil response under seismic loading [9; 10; 11; 12; 13]. The parameters for the Sig3 model were selected to match the plastic index (PI) = 0 curves of Darendeli [14] in the middle of each soil layer. The Mohr-Coulomb model was used with the Sig3 model to simulate plastic deformation, which is induced at large shear strains. The small strain damping was modeled via the Rayleigh damping formulation, which was defined as follows [15]:  
\[[C]=\alpha[M]+\beta[K] \tag{2}\]  
where [\(C\)] is the damping matrix, [\(M\)] is the mass matrix, [\(K\)] is the stiffness matrix, and \(\alpha\) and \(\beta\) are the Rayleigh coefficients that determine the frequency dependence of the damping formulation. These coefficients were defined as follows [15]:  
\[\begin{array}{l}\alpha=\frac{4\pi\xi f_{m}f_{n}}{f_{m}+f_{n}}\\ \beta=\frac{\xi}{\pi(f_{m}+f_{n})}\end{array} \tag{3}\]  
where \(\xi\) is the damping ratio. The formulation matched the target damping ratio only at the frequencies of \(f_{m}\) and \(f_{n}\). In the analyses, the 1 st and 5 th modes were used, as recommended by [PERSON] et al. [9].  
FLAC\({}^{\text{2D}}\) used the mixed discretization scheme to subdivide each quadrilateral element into two overlaid constant-strain triangle elements. It was reported that the mixed scheme accurately modeled the plastic collapse load [16]. Free-field boundary conditions were applied to the lateral boundaries to absorb reflected waves. The viscous damper proposed by [PERSON] and [PERSON] [17] was applied  
Figure 2: Shear wave velocity profiles of the rockfill and the core.  
at the lower boundary to absorb the downward propagating motion. The size of the elements was selected to propagate a minimum of 25 Hz components using the following equation:  
\[H=\frac{V_{s}}{4f_{\text{max}}} \tag{4}\]  
where \(H\) is the height of the element, \(f_{\text{max}}\) is the maximum frequency of the input motion, and \(V_{s}\) is the shear wave velocity.