5. Soils - Deformation calculation

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
F
M
M’
x’
y’
+
+
x
y
),( ),( 'yxv yxu MMu
u
Displacement vector
y
x
M
vyy uxx
M'
'
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
y
x

CALCULATION IN DEFORMATION
2. Analytic solution
Solid strain in point M
x
u
x
y
v
y
x
v
y
u
xy 
Material constitutive model
yxx
 )2(
yxy
)2( 
xyxy
General law of the equilibrium
0
i
j
ij F
x
0
x
xy
xF
yx
0
y
xyyF
xy
)21)(1(
E
)1(2
E
and : Lamé coefficients

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