Finite Element Method in Geotechnical Engineering
FINITE ELEMENT METHOD IN GEOTECHNICAL
ENGINEERING
Computational Geotechnics
Course ‘Computational Geotechnics’
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Finite Element Method in Geotechnical Engineering
Contents
Steps in the FE Method
Introduction to FEM for Deformation Analysis
Discretization of a Continuum
Elements
Strains
Stresses, Constitutive Relations
Hooke’s Law
Formulation of Stiffness Matrix
Solution of Equations
Course ‘Computational Geotechnics’
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Finite Element Method in Geotechnical Engineering
Steps in the FE Method
1. Establishment of stiffness relations for each element. Material properties
and equilibrium conditions for each element are used in this establishment.
2. Enforcement of compatibility, i.e. the elements are connected.
3. Enforcement of equilibrium conditions for the whole structure, in the present
case for the nodal points.
4. By means of 2. And 3., the system of equations is constructed for the whole
structure. This step is called assembling.
5. In order to solve the system of equations for the whole structure, the
boundary conditions are enforced.
6. Solution of the system of equations.
Course ‘Computational Geotechnics’
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Finite Element Method in Geotechnical Engineering
Introduction to FEM for Deformation Analysis
Finite Element Method:
• General method to solve boundary value problems in an approximate and
discretized way
• Often (but not only) used for deformation and stress analysis
• Division of geometry into finite element mesh
• Pre-assumed interpolation of main quantities (displacements) over elements,
based on values in points (nodes)
• Formation of (stiffness) matrix, K, and (force) vector, r
• Global solution of main quantities in nodes, d
dD

KD=R
rR
kK
Course ‘Computational Geotechnics’
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Finite Element Method in Geotechnical Engineering
Discretization of a Continuum
2D modeling:
2D cross section is divided into element
Several element types are possible (triangles and quadrilaterals)
Course ‘Computational Geotechnics’
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Finite Element Method in Geotechnical Engineering
Elements
Different types of 2D elements:
Example:
Other way of writing:
ux = N1 ux1 + N2 ux2 + N3 ux3 + N4 ux4 + N5 ux5 + N6 ux6
uy = N1 uy1 + N2 uy2 + N3 uy3 + N4 uy4 + N5 uy5 + N6 uy6
or
ux = N ux
and
u y = N uy
Course ‘Computational Geotechnics’
(N contains functions of x and y)
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Finite Element Method in Geotechnical Engineering
Strains
Strains are the derivatives of displacements. In finite elements they are
determined from the derivatives of the interpolation functions:
u x
N
 a1  2a3 x  a4 y 
ux
x
x
u
N
 yy  y  b2  2b4 x  b5 y 
uy
y
y
u u
N
N
 xy  x  y  (b1  a2 )  (a4  2b3 ) x  (2a5  b4 ) y 
ux 
uy
y
x
x
y
 xx 
or
ε  Bd
(strains composed in a vector)
(matrix B contains derivatives of N)
Course ‘Computational Geotechnics’
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Finite Element Method in Geotechnical Engineering
Stresses, Constitutive Relations
Cartesian stress tensor, usually composed in a vector:
σ  ( xx ,  yy ,  zz ,  xy ,  yz ,  yx )T
plane strain:
 yz   zx  0
(  zz is generally NOT zero!)
Stress, , are related to strain :
=C
In fact, the above relationship is used in incremental form:


σ  Cε
or
σ  Cε
C is material stiffness matrix and determining material behavior
Course ‘Computational Geotechnics’
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Finite Element Method in Geotechnical Engineering
Hooke’s Law
For simple linear elastic behavior C is based on Hooke’s law:


1 
 
1 


 
 1 
E
C

0
0
(1  2 )(1   )  0
 0
0
0

0
0
 0
0
0
0
0
0
1 
2
0
0
0
0
1
2

0
0 
0 
0 

0 
0 

1 
2

Basic parameters in Hooke’s law:
Young’s modulus E
Poisson’s ratio 
Auxiliary parameters, related to basic parameters:
Shear modulus
G
E
2(1   )
Bulk modulus
K
E
3(1  2 )
Oedometer modulus
Eoed 
E (1  )
(1  2 )(1   )
Course ‘Computational Geotechnics’
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Finite Element Method in Geotechnical Engineering
Hooke’s Law
Meaning of parameters
E
 1
 2
in axial compression
 
 3
1
in axial compression
Eoed 
 1
1
axial compression
1D compression
in 1D compression
K
p
 v
in volumetric
compression
G
 xy
 xy
in shearing
Course ‘Computational Geotechnics’
note:
 xy   xy
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Finite Element Method in Geotechnical Engineering
Hooke’s Law
Summary, Hooke’s law:


  xx 
1 




1 

yy


 
  zz 
 
 1 
E




0
0
  xy  (1  2 )(1   )  0
  yz 
 0
0
0



0
0
 0
  zx 
1
2
0
0
0
0
0
0

0
0
1
2
0

0
0    xx 
 
0    yy 
0    zz 
 
0    xy 
0    yz 
  
1    
2
  zx 
Inverse relationship:
  xx 
 1
 


 yy 
 
  zz  1  
  
  xy  E  0
  yz 
 0
 


 0
 zx 


0
0
1

0
0

1
0
0
0
0
2  2
0
0
0
0
2  2
0
0
0
0
Course ‘Computational Geotechnics’
   xx 


0    yy 
0    zz 


0    xy 
0    yz 


2  2    zx 
0
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Finite Element Method in Geotechnical Engineering
Formulation of Stiffness Matrix
Formation of element stiffness matrix Ke
K e   BT CBdV
Integration is usually performed numerically: Gauss integration
n
 pdV    p
i 1
i
(summation over sample points)
i
coefficients
integration is exact
Formation of global stiffness matrix
Assembling of element stiffness matrices in global matrix
K is often symmetric and has a band-form:
#

#
0

0
0

0
0

0
0

0

#
#
#
0
0
0
0
0
0
0
0
#
#
#
0
0
0
0
0
0
0
0
#
#
#
0
0
0
0
0
0
0
0
#
#
#
0
0
0
0
0
0
0
0
#
#
#
0
0
0
0
0
0
0
0
#
#
#
0
0
Course ‘Computational Geotechnics’
0
0
0
0
0
0
#
#
#
0
0
0
0
0
0
0
0
#
#
#
0

0
0

0
0

0
0

0
# 
# 
(# are non-zero’s)
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Finite Element Method in Geotechnical Engineering
Solution of Equations
Global system of equations:
KD  R
R
is force vector and contains loadings as nodal forces
Usually in incremental form:
KD  R
Solution:
D  K 1R
n
D   D
(i = step number)
i 1
From solution of displacements
D  d
Strains:
 εi  Bui
Stresses:
 σi  σi1  Cd
Course ‘Computational Geotechnics’
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