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# 5. Soils - Deformation calculation

publicité ```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
```