Finite Element Method in Geotechnical Engineering

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. Finite Element Method in Geotechnical Engineering Computational Geotechnics 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

3. Finite Element Method in Geotechnical Engineering Computational Geotechnics 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.

4. Finite Element Method in Geotechnical Engineering Computational Geotechnics 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

5. Finite Element Method in Geotechnical Engineering Computational Geotechnics 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

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

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

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

9. Finite Element Method in Geotechnical Engineering Computational Geotechnics 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 = Nux and uy = Nuy (N contains functions of x and y)

10. Finite Element Method in Geotechnical Engineering Computational Geotechnics 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 )

11. Finite Element Method in Geotechnical Engineering Computational Geotechnics 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

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

13. 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 Finite Element Method in Geotechnical Engineering Computational Geotechnics Hooke’s Law

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

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

16. Hooke’s Law Summary, Hooke’s law:

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

18. Finite Element Method in Geotechnical Engineering Computational Geotechnics 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

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

20. Solution of Equation Global system of equations: KD = R Ris force vector and contains loadings as nodal forces Usually in incremental form: Solution: (i = step number)

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