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 Contents Steps in the FE Method Introduction to FEM for Deformation Analysis Discretization of a Continuum Elements Strains Stresses, Constitutive Relations Hookes Law Formulation of Stiffness Matrix Solution of Equations Computational Geotechnics Finite Element Method in Geotechnical Engineering

Steps in the FE Method 1. 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. Computational Geotechnics Finite Element Method in Geotechnical Engineering Introduction to FEM for

Deformation Analysis General method to solve boundar y value problems in an approxima te and discretized way Often (but not only) used for defo rmation and stress analysis Division of geometry into finite el ement mesh Computational Geotechnics Finite Element Method in Geotechnical Engineering 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 dD KD=R rR kK Computational Geotechnics Finite Element Method in Geotechnical Engineering Discretization of a Continuum 2D modeling: Computational Geotechnics Finite Element Method in Geotechnical Engineering 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 Elements Different types of 2D elements:

Computational Geotechnics Finite Element Method in Geotechnical Engineering 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 Strains Strains are the derivatives of displacements. In finite elements they are det ermined from the derivatives of the interpolation functions: or u N xx x a1 2a3 x a4 y ux x x u N yy y b2 2b4 x b5 y uy y y u u

N N xy x y (b1 a2 ) (a4 2b3 )x (2a5 b4 )y ux uy y x x y Bd (strains composed in a vector and matrix B contains derivatives of N ) Computational Geotechnics Finite Element Method in Geotechnical Engineering Stresses, Constitutive Relations Cartesian stress tensor, usually composed in a vector: Stresses, , are related to strains : = C 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 Hookes Law For simple linear elastic behavior C is based on Hookes law: 1

1 1 E C 0 0 (1 2 )(1 ) 0 0 0 0 0 0 0 Computational Geotechnics 0 0 0 0 1 2 0 0

0 0 0 1 2 0 0 0 0 0 0 1 2 Finite Element Method in Geotechnical Engineering Hookes Law Basic parameters in Hookes law: Youngs modulus E Poissons ratio Auxiliary parameters, related to basic parameters: Shear modulus E

G 2(1 ) Bulk modulus E K 3(1 2 ) Computational Geotechnics Oedometer modulus E(1 ) E oed (1 2 )(1 ) Finite Element Method in Geotechnical Engineering Hookes Law Meaning of parameters E 1 2 in axial compression

3 1 in axial compression axial compression E oed 1D compression 1 1 in 1D compression Computational Geotechnics Finite Element Method in Geotechnical Engineering Hookes Law Meaning of parameters p K v in volumetric compression xy G

xy in shearing note: xy xy Computational Geotechnics Finite Element Method in Geotechnical Engineering Hookes Law Summary, Hookes law: xx 1 1 yy zz 1 E 0 0 xy (1 2 )(1 ) 0

yz 0 0 0 0 0 0 zx 1 2 0 0 0 0 0 0 0 0 1 2 0

0 0 0 xx yy zz 0 0 xy yz 0 1 zx 2 Hookes Law Inverse relationship: xx 1 yy zz 1

xy E 0 yz 0 zx 0 Computational Geotechnics 1 0 0 0 0 1 0 0 0 0 0 0 2 2 0

0 2 2 0 0 0 0 0 xx 0 yy 0 zz 0 xy 0 yz 2 2 zx Finite Element Method in Geotechnical Engineering Formulation of Stiffness Matrix Formation of element stiffness matrix Ke K e BT CBdV Integration is usually performed numerically: Gauss integration n pdV p i i

(summation over sample points) i1 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 Formulation of Stiffness Matrix K is often symmetric and has a band-form: # # 0 0 0 0 0 0 0 0 Computational Geotechnics

# # 0 # 0 0 0 0 0 0 0 0 0 0 0 0 # 0 0 0 # # 0 0

# # # 0 0 # # # 0 0 # # 0 0 0 # 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 # 0 # # # # 0 # 0 0 0 0 0 0 0 0 0 0 0

0 # 0 # # 0 0 0 0 0 0 0 0 # # (# are non-zeros) Finite Element Method in Geotechnical Engineering Solution of Equation Global system of equations: KD = R R is force vector and contains loadings as nodal forces Usually in incremental form: Solution:

KD R 1 D K R n D D i1 (i = step number) Solution of Equations From solution of displacement D d Strains: i Bui Stresses: i i 1 Cd Computational Geotechnics Finite Element Method in Geotechnical Engineering