Finite Element Method in Geotechnical EngineeringShort Course on Computational Geotechnics + DynamicsBoulder, ColoradoJanuary 5-8, 2004

Stein StureProfessor of Civil EngineeringUniversity of Colorado at Boulder

Contents Steps in the FE MethodIntroduction to FEM for Deformation AnalysisDiscretization of a ContinuumElementsStrainsStresses, Constitutive RelationsHookes LawFormulation of Stiffness MatrixSolution of Equations

Steps in the FE MethodEstablishment 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.

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

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, rGlobal solution of main quantities in nodes, dd DK D = Rr Rk K

Discretization of a Continuum 2D modeling:

Discretization of a Continuum2D cross section is divided into element:

Several element types are possible (triangles and quadrilaterals)

Elements Different types of 2D elements:

ElementsOther way of writing:ux = N1 ux1 + N2 ux2 + N3 ux3 + N4 ux4 + N5 ux5 + N6 ux6uy = N1 uy1 + N2 uy2 + N3 uy3 + N4 uy4 + N5 uy5 + N6 uy6orux = N ux and uy = N uy (N contains functions of x and y)Example:

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 )

Stresses, Constitutive RelationsCartesian stress tensor, usually composed in a vector:Stresses, s, are related to strains e:s = CeIn fact, the above relationship is used in incremental form:

C is material stiffness matrix and determining material behavior

Hookes LawFor simple linear elastic behavior C is based on Hookes law:

Hookes LawBasic parameters in Hookes law:Youngs modulus EPoissons ratio Auxiliary parameters, related to basic parameters:Shear modulus Oedometer modulus

Bulk modulus

Hookes LawMeaning of parameters

in axial compression

in axial compression

in 1D compressionaxial compression1D compression

Hookes LawMeaning of parameters

in volumetric compression

in shearing

note:

Hookes LawSummary, Hookes law:

Hookes LawInverse relationship:

Formulation of Stiffness MatrixFormation 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

Formulation of Stiffness MatrixK is often symmetric and has a band-form:

(# are non-zeros)

Solution of EquationGlobal system of equations:KD = RR is force vector and contains loadings as nodal forcesUsually in incremental form:Solution:

(i = step number)

Solution of EquationsFrom solution of displacement

Strains:

Stresses: