Methods for Slope Failure Induced by EQ 1 There is no analytical solution for wave equation in general boundary condition. FEM (Finite Element Method), SPH, MPM, Meshfree, etc., Most widely used in many problems Difficulty to simulate slope failure a lot of cracks, i.e., small discontinuity, which are developed to a slip line, i.e., large discontinuity DEM (Distinct/Discrete Element method) The most successful method for slope failure. Does not satisfy the equation of wave motion theoretically. Difficulty in simulation of wave propagation DEM

MPS, DEM for Slope Stability Analysis 2 Wave Propagation Collapse, failure with many crack FEM DEM MPS ? Computation Time Small Large DEM (Discrete/Distinct Element Method, Cundall 1971) Good at collapse, failure problem with many cracks Difficulty in Elastic wave propagation (not satisfy wave equation) MPS （ Moving Particle Semi-implicit/Simulation, Koshizuka et.al) MPS for fluid material (1995), MPS for solid material (2000?) MPS is mathematically derived from governing equation. Its formulation is close to that of DEM (We can use the knowledge through the experience of DEM) Few applications to geotechnical problem DEM

Most important features in formulation of MPS 3 Volumetric strain Shear and Normal strain Same as DEM New Term Normal Shear Volumetric μ μ λ Lame’s Constant difference Formulation with difference not differentiation DEM can not express wave propagation precisely, because DEM does not have the 3 rd term. MPS can be interpreted as extended DEM.

Viscous Condition and Response 4 Vs =100m/s m=1.5 Vs =150m/s m=2.0 Small amount of reflection is observed, but generally good result Viscous 2E+input Visous

Elastic Body Wave Propagation Comprehensive Check with 2-D FEM 5 3 layers slope model Particle(Element)-wise Rayleigh Damping 7%, 3%, 3% Viscous boundary condition for input Ricker Wavelet transmission and reflection, coupling

6 Boundary Condition and Material Properties Rayleigh Damping ： Characteristic frequency 1Hz, 15Hz PropertyUnitABC Unit WeightkN/m Shear ModulusMPa Poisson’s Ratio － Damping Ratio% Boundary Condition Base : Viscous Boundary, Side : Free Input Motion : Ricker Wavelet Material Property Input Motion Ricker Wavelet

7 Response Acceleration FEMMPS: r=1.0m horizontal direction RedBlue Time=0.74 sec.Red 0.2m/s 2 Response at Top Slope Acc. Time, (sec.) Acc. Time, (sec.) Horizontal DirectionVertical Direction