STABILITY ANALYSIS IN SOIL MASSES CALCULATIONS IN DEFORMATION Jean-Alain Fleurisson, CESECO-Géosciences LIMIT EQUILIBRIUM METHODS BASIS HYPOTHESIS Potential failure surface is known Instantaneous failure Maximum available shear strength of the material involved in the failure is completely and uniformaly mobilized in any point of the failure surface Evaluation of the slope stability through a Factor of Safety LIMITATIONS Deformation and failure mechanisms (over?) simplified Progressive deformation and failure mechanism is not considered Deformation or displacement are not calculated even if the slope is stable CALCULATION IN DEFORMATION 1. Problem presentation y F u x u ( x, y ) u MM ' v ( x, y ) x M y + + M’ x’ y’ Displacement vector x M y x' x u M y' y v Determine the u vector in any point of the solid body Subjected to the action of the external forces F and considering the boundary conditions CALCULATION IN DEFORMATION 2. Analytic solution Solid strain in point M u x x v y y xy u v y x Material constitutive model x ( 2 ) x y y x ( 2 ) y xy xy General law of the equilibrium ij x j Fi 0 and : Lamé coefficients E E 2(1 ) (1 )(1 2 ) x xy Fx 0 x y y y xy x Fy 0 CALCULATION IN DEFORMATION 3. Finite Element method or Finite Difference Method Transform the system of differential equations in a system of matrix equations Divide the domain in n ELEMENTS connected by a finite number of points called NODES located on their boundaries Node displacements are the unknown quantities of the problem Select specific functions in order to define the state of displacement inside the element as a function of the node displacement (only for FEM) Minimize the total potential energy in order to calculate the variation of the displacements representing the reality Systeme of 2n equations with 2n unknown quantities which are representing the node displacements Node displacements Strains Stresses FLAC - 2D Fast Lagragian Analysis of Continuous Media Finite Difference codes developped by Dr. Peter Cundall in 1996 and the ITASCA company to perform engineering applications in the field of mining, underground works, rock mechanics and research Finite Difference Method Full dynamic equation of motion Explicite time marching scheme Equilibrium equation (Movement equation) New velocities and New displacements New forces and New forces Relation Stress-strain (Constitutive equations) Hydraulic conditions: - Fixed pore pressure distribution (water table) - Flow of groundwater 2D plane strain state 2D plane stress for eleastic analysis 2D axisymmetric geometry Initial grid (slope angle 30°) Final grid (slope angle 30°) Mohr-Coulomb model (φ = 32°) Mohr-Coulomb model (φ = 32°) Mohr-Coulomb model (φ = 32°) Mohr-Coulomb model (φ = 32°) Mohr-Coulomb model (φ = 30°) Mohr-Coulomb model (φ = 30°) Mohr-Coulomb model (φ = 30°) Mohr-Coulomb model (φ = 30°) Mohr-Coulomb model (φ = 28°) Mohr-Coulomb model (φ = 28°) Mohr-Coulomb model (φ = 28°) Mohr-Coulomb model (φ = 28°) Mohr-Coulomb model (φ = 32°) Mohr-Coulomb model (φ = 32°) with watertable Mohr-Coulomb model (φ = 32°) with watertable and reservoir Mohr-Coulomb model (φ = 32°) Mohr-Coulomb model (φ = 32°) with watertable and reservoir Mohr-Coulomb model (φ = 32°) with watertable