Finite Element Method in Geotechnical Engineering

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Finite Element Method in Geotechnical Engineering

• Short Course on Computational Geotechnics
  Dynamics
• Boulder, Colorado
• January 5-8, 2004

Stein Sture Professor of Civil Engineering Univers
ity of Colorado at Boulder
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Contents

• Steps in the FE Method
• Introduction to FEM for Deformation Analysis
• Discretization of a Continuum
• Elements
• Strains
• Stresses, Constitutive Relations
• Hookes Law
• Formulation of Stiffness Matrix
• Solution of Equations

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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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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
• Division of geometry into finite element mesh

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Introduction to FEM for Deformation Analysis

• 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 ? K D R
• r ? R
• k ? K

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Discretization of a Continuum

• 2D modeling

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Discretization of a Continuum

• 2D cross section is divided into element
• Several element types are possible (triangles and
  quadrilaterals)

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Elements

• Different types of 2D elements

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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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Strains

• Strains are the derivatives of displacements.
  In finite elements they are determined from the
  derivatives of the interpolation functions
• or
• (strains composed in a vector and matrix B
  contains derivatives of N )

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Stresses, Constitutive Relations

• Cartesian stress tensor, usually composed in a
  vector
• Stresses, s, are related to strains e
• s Ce
• In fact, the above relationship is used in
  incremental form
• C is material stiffness matrix and determining
  material behavior

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Hookes Law

• For simple linear elastic behavior C is based on
  Hookes law

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Hookes Law

• Basic parameters in Hookes law
• Youngs modulus E
• Poissons ratio ?
• Auxiliary parameters, related to basic
  parameters
• Shear modulus
  Oedometer modulus
• Bulk modulus

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Hookes Law

• Meaning of parameters
• in axial compression
• in axial compression
• in 1D compression

axial compression
1D compression
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Hookes Law

• Meaning of parameters
• in volumetric compression
• in shearing
• note

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Hookes Law

• Summary, Hookes law

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Hookes Law

• Inverse relationship

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Formulation of Stiffness Matrix

• Formation of element stiffness matrix Ke
• Integration is usually performed numerically
  Gauss integration
• 
  (summation over sample
  points)
• coefficients ? and position of sample points can
  be chosen such that the integration is exact
• Formation of global stiffness matrix
• Assembling of element stiffness matrices in
  global matrix

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Formulation of Stiffness Matrix

• K is often symmetric and has a band-form
• 
  (
  are non-zeros)

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Solution of Equation

• Global system of equations
• KD R
• R is force vector and contains loadings as
  nodal forces
• Usually in incremental form
• Solution
• 
  (i step number)

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Solution of Equations

• From solution of displacement
• Strains
• Stresses