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

� Hooke’s 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 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

Computational Geotechnics Finite Element Method in Geotechnical Engineering

� 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

d ⇒⇒⇒⇒ D →→→→ K D = R

r ⇒⇒⇒⇒ R

k ⇒⇒⇒⇒ K

Introduction to FEM for

Deformation Analysis

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

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)

Example:

Computational Geotechnics Finite Element Method in Geotechnical Engineering

Strains

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

or

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

εxx

=∂u

x

∂x= a

1+ 2a

3x + a

4y =

∂N∂x

ux

εyy =∂u

y

∂y= b2 + 2b4x + b5y =

∂N∂y

uy

γxy

=∂u

x

∂y+

∂uy

∂x= (b

1+ a

2)+ (a

4+ 2b

3)x + (2a

5+ b

4)y =

∂N∂x

ux

+∂N∂y

uy

ε =Bd

Computational Geotechnics Finite Element Method in Geotechnical Engineering

Stresses, Constitutive Relations

Cartesian stress tensor, usually

composed in a vector:

Stresses, σσσσ , are related to strains εεεε:

σσσσ = Cεεεε

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

Hooke’s Law

For simple linear elastic behavior C is based on

Hooke’s law:

C =E

(1− 2ν)(1+ ν )

1− ν ν ν 0 0 0

ν 1−ν ν 0 0 0

ν ν 1− ν 0 0 0

0 0 0 12

−ν 0 0

0 0 0 0 12

− ν 0

0 0 0 0 0 1

2− ν

Computational Geotechnics Finite Element Method in Geotechnical Engineering

Hooke’s Law

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

Poisson’s ratio ν

Auxiliary parameters, related to basic parameters:

Shear modulus Oedometer modulus

Bulk modulus

G =E

2(1+ ν )

K =E

3(1− 2ν )

Eoed =E (1−ν )

(1− 2ν )(1+ ν )

Computational Geotechnics Finite Element Method in Geotechnical Engineering

Hooke’s Law

Meaning of parameters

in axial compression

in axial compression

in 1D compression

E =∂σ1

∂σ2

ν = −∂ε3∂ε1

Eoed =∂σ1

∂ε1

Computational Geotechnics Finite Element Method in Geotechnical Engineering

axial compression 1D compression

Hooke’s Law

Meaning of parameters

in volumetric compression

in shearing

note:

K =∂p∂εv

G =∂σ

xy

∂γxy

σxy

≡ τxy

Computational Geotechnics Finite Element Method in Geotechnical Engineering

Hooke’s Law

Summary, Hooke’s law:

σ xx

σ yy

σzz

σxy

σyz

σ zx

=E

(1− 2ν )(1+ ν )

1− ν ν ν 0 0 0

ν 1− ν ν 0 0 0

ν ν 1− ν 0 0 0

0 0 0 1

2− ν 0 0

0 0 0 0 12

− ν 0

0 0 0 0 0 12

−ν

εxxεyyεzz

εxy

εyz

εzx

Hooke’s Law

Inverse relationship:

εxxεyy

εzzεxyεyz

εzx

=1

E

1 −ν −ν 0 0 0

−ν 1 −ν 0 0 0

−ν −ν 1 0 0 0

0 0 0 2+ 2ν 0 0

0 0 0 0 2 + 2ν 0

0 0 0 0 0 2+ 2ν

σxx

σyy

σ zz

σxy

σyz

σ zx

Computational Geotechnics Finite Element Method in Geotechnical Engineering

Formulation of Stiffness Matrix

Formation of element stiffness matrix Ke

Integration is usually performed numerically: Gauss integration

(summation over sample points)

coefficients α 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

∫= dVTeCBBK

pdV = α ipii=1

n

∑∫

Computational Geotechnics Finite Element Method in Geotechnical Engineering

Formulation of Stiffness Matrix

K is often symmetric and has a band-form:

(# 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 Equation

Global system of equations:

KD = R

R is force vector and contains loadings as nodal forces

Usually in incremental form:

Solution:

(i = step number)

K∆D = ∆R

∆D = K−1∆R

D = ∆Di=1

n

∑

Solution of Equations

From solution of displacement

Strains:

Stresses:

∆D⇒ ∆d

→ ∆εi =B∆ui

→σ i = σ i−1 +C∆d

Computational Geotechnics Finite Element Method in Geotechnical Engineering