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 RelationsHooke’s 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 presentcase 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 Þ D ® K D = Rr Þ Rk Þ 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 determinedfrom 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 b x a b yy x x yeeg� �= = + + =� ���= = + + =� ���� �= + = + + + + + = +� � � �NuNuN Nu uor=ε 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 yxs s s s s s=σplane strain: 0yz zxs s= =(zzs is generally NOT zero!)Stress, , are related to strain : = C In fact, the above relationship is used in incremental form:� �=σ Cε or D = Dσ C εC is material stiffness matrix and determining material behaviorCourse ‘Computational Geotechnics’ 8Finite Element Method in Geotechnical EngineeringHooke’s LawFor simple linear elastic behavior C is based on Hooke’s 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 0En n nn n nn n nnn nnn-� �� �-� �� �-=� �-- +� �� �-� �-� �� �CBasic parameters in Hooke’s law:Young’s modulus EPoisson’s ratio Auxiliary parameters, related to basic parameters:Shear modulus2(1 )EGn=+Bulk modulus3(1 2 )EKn=-Oedometer modulus(1 )(1 2 )(1 )oedEEnn n-=- +Course ‘Computational Geotechnics’ 9Finite Element Method in Geotechnical EngineeringHooke’s LawMeaning of parameters12Ess�=�in axial compression axial compression 1D compression31ene�=-�in axial compression11oedEse�=�in 1D compressionvpKe�=�in volumetric compressionxyxyGsg�=�in shearingnote:xy xys t�Course ‘Computational Geotechnics’ 10Finite Element Method in Geotechnical EngineeringHooke’s LawSummary, Hooke’s 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 zxEs en n ns en n ns en n ns gnn ns gns gn-� � � �� �� � � �� �-� � � �� �� � � �� �-=� � � �� �-- +� � � �� �� � � �� �-� � � �� �� �� � � �-� �� � � �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 zxEe sn ne sn ne sn ne sne sne sn- -� � � �� �� � � �� �- -� � � �� �� � � �� �- -=� � � �� �+� � � �� �� � � �� �+� � � �� �� �� � � �+� �� � � �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 pa==��(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