1. Aleatory Uncertainty in Non-linear Behavior Observed in MPS Simulation Ikumasa Yoshida Tokyo City University

2. Collapse Ratio =Ratio of to all particles Collapse of Slope due to Gravity 0.0m - 0.1m 0.1m - 0.5m 0.5m or more Displacement in 1G 0.0 2.0 4.0 6.0 8.0 10.0 Time (sec.) Gravity (m/s 2 ) 1G=9.8 4 sec. 5 sec. 6 sec. 10 sec. 2

3. Deformation in early stage, weak nonlinear Disp. 0.00m ～ 0.01m (1G) 0.01m ～ 0.02m more than 0.02m Magnification 100.0 1 sec. 3 Nonlinearity is still weak though failure progresses at the left boundary. No magnification (1.0) 0.0m ～ 0.1m 0.1m ～ 0.5m 0.5m Strong nonlinearity near the slope 2 sec. 4 sec. Vertical disp. at top

4. 4 2m 60m 20m 10m30m 22m Material Property Young’s Modulus [E 0 ] 50000 kPa Coefficient [C E ] 500 kPa/m Poisson’s Ratio 0.35 Damping Ratio 0.05 Cohesion 80.0 kPa Internal Friction 30 (degree) d : depth Uncertainty due to Property perturbation E 0 Young’s Modulus C E Coefficient Case 10.10.4 Case 21.04.0 Case 310.040.0 Surface COV of Young’s Modulus (%)

5. Case1 Case2 Case3 small medium large (uncertainty) 5 Uncertainty of Response Standard deviation of the vertical displacement Vertical displacement of particle at slope top 1 sec. proportional to COV of Young’s modulus 2 sec. 4 sec. Almost same St.Dev. despite of Young’s modulus Collapse Ratio =Ratio of “>0.5m” Almost same St.Dev. =0.27 Rosenblueth method to save computation time, MCS (size,10,20) is also performed in a specific case. Almost same result was obtained. 1.0 sec. 2.0 sec. 4.0 sec. Collapse ratio

6. Uncertainty of Response Standard deviation of the vertical displacement DEM If you place the particles regularly, the regularity affect the failure phenomenon strongly. Packing (initial placement of particles) is important. Same goes for MPS method. Several initial models are prepared and standard deviations of the response are estimated. (Particle size does not affect?) Collapse Ratio =Ratio of “>0.5m” Almost same St.Dev. =0.27 Rosenblueth method to save computation time, MCS (size,10,20) is also performed in a specific case. Almost same result was obtained. Initial Case (Property) Model 1 2 3 Input uncertainty Small Large Standard Deviation Collapse Ratio 6

7. Computational Environment Treatment of significant digit depends on compiler, its option or PC, which leads to significant difference in nonlinear response. Displacement in early stage (linear response) is same irrespective of compliers Compiler A Compiler B 7 Collapse Ratio 0.27 Collapse Ratio 0.24 All input data, program are same except Compiler