FEM in Geotech Engineering

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Finite Element Finite Element Method in Method in Geotechnical Geotechnical Engineering Engineering Short Course on Computational Geotechnics + Dynamics Short Course on Computational Geotechnics + Dynamics Boulder, Colorado Boulder, Colorado January 5-8, 2004 January 5-8, 2004 Stein Sture Professor of Civil Engineering University of Colorado at Boulder

Transcript of FEM in Geotech Engineering

Finite Element Method in Finite Element Method in Geotechnical EngineeringGeotechnical Engineering

Short Course on Computational Geotechnics + DynamicsShort Course on Computational Geotechnics + DynamicsBoulder, ColoradoBoulder, ColoradoJanuary 5-8, 2004January 5-8, 2004

Stein StureProfessor of Civil EngineeringUniversity of Colorado at Boulder

Contents Contents

Steps in the FE MethodSteps in the FE Method Introduction to FEM for Deformation AnalysisIntroduction to FEM for Deformation Analysis Discretization of a ContinuumDiscretization of a Continuum ElementsElements StrainsStrains Stresses, Constitutive RelationsStresses, Constitutive Relations Hooke’s LawHooke’s Law Formulation of Stiffness MatrixFormulation of Stiffness Matrix Solution of EquationsSolution of Equations

Computational Geotechnics Finite Element Method in Geotechnical Engineering

Steps in the FE MethodSteps in the FE Method1.1. Establishment of stiffness relations for each element. Material Establishment of stiffness relations for each element. Material

properties and equilibrium conditions for each element are used in this properties and equilibrium conditions for each element are used in this establishment.establishment.

2.2. Enforcement of compatibility, i.e. the elements are connected.Enforcement of compatibility, i.e. the elements are connected.

3.3. Enforcement of equilibrium conditions for the whole structure, in the Enforcement of equilibrium conditions for the whole structure, in the present case for the nodal points.present case for the nodal points.

4.4. By means of 2. And 3. the system of equations is constructed for the By means of 2. And 3. the system of equations is constructed for the whole structure. This step is called assembling.whole structure. This step is called assembling.

5.5. In order to solve the system of equations for the whole structure, the In order to solve the system of equations for the whole structure, the boundary conditions are enforced.boundary conditions are enforced.

6.6. Solution of the system of equations.Solution of the system of equations.

Computational Geotechnics Finite Element Method in Geotechnical Engineering

Introduction to FEM for Introduction to FEM for Deformation Analysis Deformation Analysis

General method to solve boundarGeneral method to solve boundary value problems in an approximay value problems in an approximate and discretized wayte and discretized way

Often (but not only) used for defoOften (but not only) used for deformation and stress analysisrmation and stress analysis

Division of geometry into finite elDivision of geometry into finite element meshement mesh

Computational Geotechnics Finite Element Method in Geotechnical Engineering

Pre-assumed interpolation of main Pre-assumed interpolation of main quantities (displacements) over quantities (displacements) over elements, based on values in points elements, based on values in points (nodes)(nodes)

Formation of (stiffness) matrix, Formation of (stiffness) matrix, KK, , and (force) vector, and (force) vector, rr

Global solution of main quantities Global solution of main quantities in nodes, in nodes, dd

d d D D K D = RK D = R

r r R R

k k K K

Introduction to FEM for Introduction to FEM for Deformation AnalysisDeformation Analysis

Computational Geotechnics Finite Element Method in Geotechnical Engineering

Discretization of a Continuum Discretization of a Continuum

2D modeling: 2D modeling:

Computational Geotechnics Finite Element Method in Geotechnical Engineering

Discretization of a ContinuumDiscretization of a Continuum

2D cross section is divided into element:2D cross section is divided into element:

Several element types are possible (triangles and quadrilaterals)Several element types are possible (triangles and quadrilaterals)

Computational Geotechnics Finite Element Method in Geotechnical Engineering

Elements Elements

Different types of 2D elements:Different types of 2D elements:

Computational Geotechnics Finite Element Method in Geotechnical Engineering

ElementsElements

Other way of writing:Other way of writing:

uuxx = N = N11 u ux1x1 + N + N22 u ux2x2 + N + N33 u ux3x3 + N + N44 u ux4x4 + N + N55 u ux5x5 + N + N66 u ux6x6

uuyy = N = N11 u uy1y1 + N + N22 u uy2y2 + N + N33 u uy3y3 + N + N44 u uy4y4 + N + N55 uy uy55 + N + N66 u uy6y6

oror

uuxx = = NN uux x and u and uyy = = NN uuyy ( (NN contains functions of x and y) contains functions of x and y)

Example:

Computational Geotechnics Finite Element Method in Geotechnical Engineering

Strains Strains

Strains are the derivatives of displacements. In finite elements they are detStrains are the derivatives of displacements. In finite elements they are determined from the derivatives of the interpolation functions:ermined from the derivatives of the interpolation functions:

oror

(strains composed in a vector and matrix B contains derivatives of N )(strains composed in a vector and matrix B contains derivatives of N )

xx ux

xa1 2a3x a4 y

N

xux

yy uy

yb2 2b4 x b5y

N

yuy

xy ux

y

uy

x(b1 a2) (a4 2b3)x (2a5 b4 )y

N

xux

N

yuy

Bd

Computational Geotechnics Finite Element Method in Geotechnical Engineering

Stresses, Constitutive RelationsStresses, Constitutive Relations

Cartesian stress tensor, usually Cartesian stress tensor, usually composed in a vector:composed in a vector:

Stresses, Stresses, , are related to strains , are related to strains ::

= = CC

In fact, the above relationship is used In fact, the above relationship is used in incremental form:in incremental form:

C is material stiffness matrix and C is material stiffness matrix and determining material behaviordetermining material behavior

Computational Geotechnics Finite Element Method in Geotechnical Engineering

Hooke’s LawHooke’s Law

For simple linear elastic behavior C is based on For simple linear elastic behavior C is based on Hooke’s law:Hooke’s law:

C E

(1 2)(1 )

1 0 0 0

1 0 0 0

1 0 0 0

0 0 0 12 0 0

0 0 0 0 12 0

0 0 0 0 0 12

Computational Geotechnics Finite Element Method in Geotechnical Engineering

Hooke’s LawHooke’s Law

Basic parameters in Hooke’s law:Basic parameters in Hooke’s law:Young’s modulus Young’s modulus EE

Poisson’s ratio Poisson’s ratio

Auxiliary parameters, related to basic parameters:Auxiliary parameters, related to basic parameters:Shear modulus Oedometer modulusShear modulus Oedometer modulus

Bulk modulusBulk modulus

G E

2(1)

K E

3(1 2 )

Eoed E(1 )

(1 2 )(1 )

Computational Geotechnics Finite Element Method in Geotechnical Engineering

Hooke’s LawHooke’s Law

Meaning of parametersMeaning of parameters

in axial compressionin axial compression

in axial compressionin axial compression

in 1D compressionin 1D compression

E 1

2

3

1

Eoed 1

1

Computational Geotechnics Finite Element Method in Geotechnical Engineering

axial compression 1D compression

Hooke’s LawHooke’s Law

Meaning of parametersMeaning of parameters

in volumetric compressionin volumetric compression

in shearingin shearing

note:note:

K p

v

G xy

xy

xy xy

Computational Geotechnics Finite Element Method in Geotechnical Engineering

Hooke’s LawHooke’s Law

Summary, Hooke’s law:Summary, Hooke’s law:

xx

yy

zz

xy

yz

zx

E

(1 2 )(1 )

1 0 0 0

1 0 0 0

1 0 0 0

0 0 0 12 0 0

0 0 0 0 12 0

0 0 0 0 0 12

xx

yy

zz

xy

yz

zx

Hooke’s LawHooke’s Law

Inverse relationship:Inverse relationship:

xx

yy

zz

xy

yz

zx

1

E

1 0 0 0

1 0 0 0

1 0 0 0

0 0 0 2 2 0 0

0 0 0 0 2 2 0

0 0 0 0 0 2 2

xx

yy

zz

xy

yz

zx

Computational Geotechnics Finite Element Method in Geotechnical Engineering

Formulation of Stiffness MatrixFormulation of Stiffness Matrix

Formation of element stiffness matrix Formation of element stiffness matrix KKee

Integration is usually performed numerically: Gauss integrationIntegration is usually performed numerically: Gauss integration

(summation over sample points)(summation over sample points)

coefficients coefficients and position of sample points can be chosen such that the integration is exact and position of sample points can be chosen such that the integration is exact

Formation of global stiffness matrixFormation of global stiffness matrix

Assembling of element stiffness matrices in global matrixAssembling of element stiffness matrices in global matrix

dVTe CBBK

pdV i pi

i1

n

Computational Geotechnics Finite Element Method in Geotechnical Engineering

Formulation of Stiffness MatrixFormulation of Stiffness Matrix

KK is often symmetric and has a band-form: is often symmetric and has a band-form:

(# are non-zero’s)(# are non-zero’s)

# # 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 # # #

0 0 0 0 0 0 0 0 # #

Computational Geotechnics Finite Element Method in Geotechnical Engineering

Solution of EquationSolution of Equation

Global system of equations:Global system of equations:KDKD = = RR

RR is force vector and contains loadings as nodal forcesis force vector and contains loadings as nodal forces

Usually in incremental form:Usually in incremental form:Solution:Solution:

((i i = step number)= step number)

KD R

D K 1R

D Di1

n

Solution of EquationsSolution of Equations

From solution of displacementFrom solution of displacement

Strains:Strains:

Stresses:Stresses:

D d

i Bui

i i 1 Cd

Computational Geotechnics Finite Element Method in Geotechnical Engineering