Transcript Finite Element Method in Geotechnical Engineering

Finite Element Method in
Geotechnical Engineering
Short Course on Computational Geotechnics + Dynamics
Boulder, Colorado
January 5-8, 2004
Stein Sture
Professor of Civil Engineering
University of Colorado at Boulder
Contents
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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
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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.
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Finite Element Method in Geotechnical Engineering
Introduction to FEM for
Deformation Analysis

General method to solve boundary
value problems in an approximate
and discretized way

Often (but not only) used for
deformation and stress analysis
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Division of geometry into finite
element mesh
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Finite Element Method in Geotechnical Engineering
Introduction to FEM for
Deformation Analysis
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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
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Finite Element Method in Geotechnical Engineering
Discretization of a Continuum

2D modeling:
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Finite Element Method in Geotechnical Engineering
Discretization of a Continuum

2D cross section is divided into element:
Several element types are possible (triangles and quadrilaterals)
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Finite Element Method in Geotechnical Engineering
Elements

Different types of 2D elements:
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Finite Element Method in Geotechnical Engineering
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 uy = N uy (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:
ux
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

  Bd
(strains composed in a vector and matrix B contains derivatives of N )
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Finite Element Method in Geotechnical Engineering
Stresses, Constitutive Relations
Cartesian stress tensor, usually
composed in a vector:
Stresses, s, are related to strains :
s = C
In fact, the above relationship is used
in incremental form:
C is material stiffness matrix and
determining material behavior
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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
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1
2
0
0
0
0
0
0

0
0
0
1
2

0
0 

0 
0 

0 
0 

1



2
Finite Element Method in Geotechnical Engineering
Hooke’s Law
Basic parameters in Hooke’s law:
Young’s modulus E
Poisson’s ratio 
Auxiliary parameters, related to basic parameters:
Shear modulus
E
G
2(1  )

Bulk modulus
E
K
3(1 2 )
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
Oedometer modulus
E(1  )
E oed 
(1 2 )(1  )

Finite Element Method in Geotechnical Engineering

Hooke’s Law
Meaning of parameters
s 1
E
s 2
in axial compression
 
3
1
in axial compression
E oed 
s 1
1
axial compression
1D compression
in 1D compression
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Finite Element Method in Geotechnical Engineering


Hooke’s Law
Meaning of parameters
p
K
v
in volumetric compression
s xy
G
 xy
in shearing
note:
s xy   xy
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Finite Element Method in Geotechnical Engineering
Hooke’s Law
Summary, Hooke’s law:
s xx 
1 


 

s
1 

 yy 
 
s zz 
 
 1 
E
 

s
0
0
(1
2

)(1

)
 xy 
 0
s yz 
 0
0
0
 

s
0
0
 0
 zx 
1
2
0
0
0
0
0
0

0
0
0
1
2

0
0 xx 
 
0 yy 
0 zz 
 
0 xy 
0 yz 
 
1


zx 
2
Hooke’s Law
Inverse relationship:
xx 
1
 


 yy 

zz  1 
  
xy  E 0
yz 
0
 

0
zx 
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

0
0
1

0
0

1
0
0
0
0
2  2
0
0
0
0
2  2
0
0
0
0
0 s xx 
 
0 s yy 
0 s zz 
 
0 s xy 
0 s yz 
s 
2  2  zx 
Finite Element Method in Geotechnical Engineering
Formulation of Stiffness Matrix
Formation of element stiffness matrix Ke
K e   BT CB dV
Integration is usually performed numerically: Gauss integration
n
 pdV   p
i
i
(summation over sample points)
i1
coefficients  and position of sample points can be chosen such that the integration is exact
Formation of global stiffness matrix
 of element stiffness matrices in global matrix
Assembling
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Finite Element Method in Geotechnical Engineering
Formulation of Stiffness Matrix
K is often symmetric and has a band-form:
#

#
0

0
0

0
0
0

0
0

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# 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 0
0
0 0
0 0
#
0

0
0

0
0

0
0
0

#
#

(# are non-zero’s)
Finite Element Method in Geotechnical Engineering
Solution of Equation
Global system of equations:
KD = R
R is force vector and contains loadings as nodal forces
Usually in incremental form:
Solution:
KD R
1
D  K R
n
D  
D
i1
(i = step number)
Solution of Equations
From solution of displacement
Dd
Strains:

 i  Bui
Stresses:  s i  s i1  Cd

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Finite Element Method in Geotechnical Engineering