Finite Element Method in Geotechnical EngineeringFINITE ELEMENT METHOD IN GEOTECHNICAL ENGINEERINGComputational GeotechnicsCourse Computational Geotechnics 1Finite Element Method in Geotechnical EngineeringContentsSteps in the FE MethodIntroduction to FEM for Deformation AnalysisDiscretization of a ContinuumElementsStrainsStresses, Constitutive RelationsHookes LawFormulation of Stiffness MatrixSolution of EquationsCourse Computational Geotechnics 2Finite Element Method in Geotechnical EngineeringSteps in the FE Method1. 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.Course Computational Geotechnics 3Finite Element Method in Geotechnical EngineeringIntroduction to FEM for Deformation AnalysisFinite Element Method: 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 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, dd D K D = Rr Rk KCourse Computational Geotechnics 4Finite Element Method in Geotechnical EngineeringDiscretization of a Continuum2D modeling:2D cross section is divided into elementSeveral element types are possible (triangles and quadrilaterals)Course Computational Geotechnics 5Finite Element Method in Geotechnical EngineeringElementsDifferent types of 2D elements:Example:Other 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)Course Computational Geotechnics 6Finite Element Method in Geotechnical EngineeringStrainsStrains are the derivatives of displacements.In finite elements they are determined from the derivatives of the interpolation functions:1 3 42 4 51 2 4 3 5 422( ) ( 2 ) (2 )xxx xyyy yyxxy x yua a x a yx xub b x b yy yuub a a bx a b yy x x y + + + + + + + + + + + NuNuN Nu uor Bd(strains composed in a vector)(matrix B contains derivatives of N)Course Computational Geotechnics 7Finite Element Method in Geotechnical EngineeringStresses, Constitutive RelationsCartesian stress tensor, usually composed in a vector:T( , , , , , )xx yy zz xy yz yx plane strain: 0yz zx (zzis generally NOT zero!)Stress, , are related to strain := C In fact, the above relationship is used in incremental form: Cor C C is material stiffness matrix and determining material behaviorCourse Computational Geotechnics 8Finite Element Method in Geotechnical EngineeringHookes LawFor simple linear elastic behavior C is based on Hookes law:1212121 0 0 01 0 0 01 0 0 00 0 0 0 0 (1 2 )(1 )0 0 0 0 00 0 0 0 0E ] ] ] ] ] + ] ] ] ] ]CBasic parameters in Hookes law:Youngs modulus EPoissons ratio Auxiliary parameters, related to basic parameters:Shear modulus2(1 )EG+Bulk modulus3(1 2 )EKOedometer modulus(1 )(1 2 )(1 )oedEE +Course Computational Geotechnics 9Finite Element Method in Geotechnical EngineeringHookes LawMeaning of parameters12Ein axial compression

axial compression 1D compression31 in axial compression11oedEin 1D compressionvpKin volumetric compressionxyxyGin shearingnote:xy xy Course Computational Geotechnics 10Finite Element Method in Geotechnical EngineeringHookes LawSummary, Hookes law:1212121 0 0 01 0 0 01 0 0 00 0 0 0 0 (1 2 )(1 )0 0 0 0 00 0 0 0 0xx xxyy yyzz zzxy xyyz yzzx zxE | ` | ` | ` + . , . , . ,Inverse relationship:1 0 0 01 0 0 01 0 0 010 0 0 2 2 0 00 0 0 0 2 2 00 0 0 0 0 2 2xx xxyy yyzz zzxy xyyz yzzx zxE | ` | ` | ` + + +. , . , . ,Course Computational Geotechnics 11Finite Element Method in Geotechnical EngineeringFormulation of Stiffness MatrixFormation of element stiffness matrix Kee TdV K B CBIntegration is usually performed numerically: Gauss integration1ni iipdV p (summation over sample points)coefficients and position of sample points can be chosen such that the integration is exactFormation of global stiffness matrixAssembling of element stiffness matrices in global matrixK is often symmetric and has a band-form:# # 0 0 0 0 0 0 0 0# # # 0 0 0 0 0 0 00 # # # 0 0 0 0 0 00 0 # # # 0 0 0 0 00 0 0 # # # 0 0 0 00 0 0 0 # # # 0 0 00 0 0 0 0 # # # 0 00 0 0 0 0 0 # # # 00 0 0 0 0 0 0 # # #0 0 0 0 0 0 0 0 # #| ` . ,(# are non-zeros)Course Computational Geotechnics 12Finite Element Method in Geotechnical EngineeringSolution of EquationsGlobal system of equations: KD RR is force vector and contains loadings as nodal forcesUsually in incremental form: K D RSolution:11ni D K RD D(i = step number)From solution of displacements D dStrains:i i B uStresses:1 i i + C dCourse Computational Geotechnics 13