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Finite Element Method in Slope

Stability Analysis

Introduction to FEM

Slope Sta bil ity Course, 20 12 -I I

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Contents Steps in the FE Method

FEM for Deformation Analysis

Discretization of a Continuum

Elements

Strains

Stresses

Constitutive Relations

Hookes Law

Formulation of Stiffness Matrix

Solution of Equations

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

Material properties and equilibrium conditions for eachelement are used in this establishment.

2. Enforcement of compatibility. i.e. the elements areconnected.

3. Enforcement of equilibrium conditions for the wholestructure, in the present case for the nodal points.

4. By means of 2 and 3, the system of equations is constructedfor the whole structure. This step is called assembling.

5. In order to solve the system of equations for the wholestructure, the boundary conditions are enforced.

6. Solution of the system of equations.

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FEM for Deformation Analysis

General method to solve boundary value problems inan approximate and discretized way.

Often (but not only) used for deformation and stressanalysis.

Division of geometry into finite element mesh.

geometry mesh

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FEM for Deformation Analysis

Pre-assumed interpolation of main quantities

(displacements) over elements, based on values in

points (nodes).

254

2

3210, yaxyaxayaxaayxu

50 aa : determined by nodal values

interpolation function: element

x

y

xuyu

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FEM for Deformation Analysis

Formation of (stiffness) matrix and (force) vector.

K

r

: Stiffness matrix

: Force vector

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FEM for Deformation Analysis

Global solution of main quantities in nodes.

Kk

RrDd

RDK

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Discretization of a Continuum 2D modeling:

Plane Strain Axi-symmetry

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Discretization of a Continuum 2D cross section is divided into element:

Several element types are possible (triangles and quadrilaterals)

Local refinement around wall

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Elements Different types of 2D elements:

(a) triangular elements

(b) quadrangular elements

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ElementsExample:

2

54

2

3210

254

23210

ybxybxbybxbbu

yaxyaxayaxaau

y

x

50

50 ,

bb

aa

: are determined by nodal values

interpolation function:

x

y

xuyu

element

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ElementsExample:

interpolation function (other way of writing):

665544332211

665544332211

yyyyyyy

xxxxxxx

uNuNuNuNuNuNu

uNuNuNuNuNuNu

or:

yy

xx

Nuu

Nuu

N : contains functions of x and y

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Strains Strains are the derivatives of displacements.

x

u

y

u

y

u

x

u yxxy

y

yyx

xx

,,

In finite elements they are determined from thederivatives of the interpolation functions:

yxxy

yyy

xxx

yxybaxbaab

yybxbb

xyaxaa

uN

uN

u

N

uN

)2()2()(

2

2

453421

542

431

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Strains (Cont.)

Or:

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

Bd

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Stresses Cartesian stress tensor, usually composed in a vector:

Tzxyzxyzzyyxx

Plane strain:

0 zxyz

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Constitutive Relations Stresses are related to strains:

C

In fact, the above relationship is used in incrementalform:

C

C : is material stiffness matrix and determining material behavior

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Hookes Law For simple linear elastic behavior C is based on

Hookes 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

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Hookes Law Basic parameters in Hookes law:

G E

2(1 )

KE

3(1 2)

Eoed

E(1 )

(1 2)(1 )

Bulk modulus

E

: Youngs modulus

: Poissons ratio

Auxiliary parameters, related to basic parameters:

Shear modulus Oedometer modulus

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Hookes Law

E

1

2

3

1

axial compression

Meaning of parameters

in axial compression:

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Hookes Law

Eoed 1

1

1D compression

Meaning of parameters

in 1D compression:

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Hookes Law

K p

v

Meaning of parameters

in volumetric compression:

volumetric compression

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Hookes Law

G

xy

xy

xy xy

Meaning of parameters

in shearing:

note:

shearing

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Hookes Law Summary, Hookes 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

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Formulation of Stiffness Matrix

dVTCBBk

Formation of element stiffness matrix:

Formation of global stiffness matrix: Assembling ofelement stiffness matrices in global matrix.

Kk

Integration is usually performed numerically: Gaussintegration.

Global matrix is often symmetric and has a band-form

(non-zeros values).

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Solution of Equations Global system of equations:

RDK

R : is force vector and contains loadings as nodal forces

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Solution of Equations

(i = step number)

RDK

D K1R

D Di1

n

Usually in incremental form:

Solution:

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Solution of Equations

Strains:

D d

i Bui

i

i1 Cd

From solution of displacement

Stresses:

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References Lectures notes in Course on Computational Geotechnics &

Dynamics, August 2003, Boulder, Colorado.

Lectures notes in Course on Computational Geotechnics, October

2007, Rio de Janeiro Brazil.

Potts D.M. & Zdravkovi L.T. (1999), Finite element analysis in

geotechnical engineering: Theory, Thomas Telford, London.

Potts D.M. & Zdravkovi L.T. (2001), Finite element analysis in

geotechnical engineering: Application, Thomas Telford, London.

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