1. 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

2. 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
Computational Geotechnics
Finite Element Method in Geotechnical Engineering

3. Steps in the FE Method
Establishment of stiffness relations for each element. Material properties and equilibrium conditions for each element are used in this establishment.
Enforcement of compatibility, i.e. the elements are connected.
Enforcement of equilibrium conditions for the whole structure, in the present case for the nodal points.
By means of 2. And 3. the system of equations is constructed for the whole structure. This step is called assembling.
In order to solve the system of equations for the whole structure, the boundary conditions are enforced.
Solution of the system of equations.
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Finite Element Method in Geotechnical Engineering

4. 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
Computational Geotechnics
Finite Element Method in Geotechnical Engineering

5. 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
Computational Geotechnics
Finite Element Method in Geotechnical Engineering

6. Discretization of a Continuum
2D modeling:
Computational Geotechnics
Finite Element Method in Geotechnical Engineering

7. Discretization of a Continuum
2D cross section is divided into element:
Several element types are possible (triangles and quadrilaterals)
Computational Geotechnics
Finite Element Method in Geotechnical Engineering

8. Elements Different types of 2D elements: Computational Geotechnics
Finite Element Method in Geotechnical Engineering

9. 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)
Computational Geotechnics
Finite Element Method in Geotechnical Engineering

10. 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 )
Computational Geotechnics
Finite Element Method in Geotechnical Engineering

11. 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
Computational Geotechnics
Finite Element Method in Geotechnical Engineering

12. Hooke’s Law
For simple linear elastic behavior C is based on Hooke’s law:
Computational Geotechnics
Finite Element Method in Geotechnical Engineering

13. Hooke’s Law Basic parameters in Hooke’s law:
Young’s modulus E
Poisson’s ratio 
Auxiliary parameters, related to basic parameters:
Shear modulus Oedometer modulus
Bulk modulus
Computational Geotechnics
Finite Element Method in Geotechnical Engineering

14. Hooke’s Law Meaning of parameters in axial compression
in 1D compression
axial compression
1D compression
Computational Geotechnics
Finite Element Method in Geotechnical Engineering

15. Hooke’s Law Meaning of parameters in volumetric compression
in shearing
note:
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Finite Element Method in Geotechnical Engineering

16. Hooke’s Law
Summary, Hooke’s law:

17. Hooke’s Law Inverse relationship: Computational Geotechnics
Finite Element Method in Geotechnical Engineering

18. 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
Computational Geotechnics
Finite Element Method in Geotechnical Engineering

19. Formulation of Stiffness Matrix
K is often symmetric and has a band-form:
(# are non-zero’s)
Computational Geotechnics
Finite Element Method in Geotechnical Engineering

20. 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)

21. Solution of Equations From solution of displacement Strains: Stresses:
Computational Geotechnics
Finite Element Method in Geotechnical Engineering