FEM is about to Finite element method. In this it is described that how FEM is done and what are the steps which we have to follow for fully FEA. Finite Element Analysis is one of the most important analysis which is used in various field.
1. Finite Element Method inFinite Element Method in
Geotechnical EngineeringGeotechnical Engineering
Short Course on Computational Geotechnics + DynamicsShort Course on Computational Geotechnics + Dynamics
Boulder, ColoradoBoulder, Colorado
January 5-8, 2004January 5-8, 2004
Stein Sture
Professor of Civil Engineering
University of Colorado at Boulder
2. ContentsContents
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
3. Steps in the FE MethodSteps in the FE Method
1.1. Establishment of stiffness relations for each element. MaterialEstablishment of stiffness relations for each element. Material
properties and equilibrium conditions for each element are used in thisproperties 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 theEnforcement 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 theBy 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, theIn 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
4. Introduction to FEM forIntroduction to FEM for
Deformation AnalysisDeformation Analysis
General method to solveGeneral method to solve
boundary value problems in anboundary value problems in an
approximate and discretized wayapproximate and discretized way
Often (but not only) used forOften (but not only) used for
deformation and stress analysisdeformation and stress analysis
Division of geometry into finiteDivision of geometry into finite
element meshelement mesh
Computational Geotechnics Finite Element Method in Geotechnical Engineering
5. Pre-assumed interpolation of mainPre-assumed interpolation of main
quantities (displacements) overquantities (displacements) over
elements, based on values inelements, based on values in
points (nodes)points (nodes)
Formation of (stiffness) matrix,Formation of (stiffness) matrix,
KK, and (force) vector,, and (force) vector, rr
Global solution of main quantitiesGlobal solution of main quantities
in nodes,in nodes, dd
dd ⇒⇒ DD →→ K D = RK D = R
rr ⇒⇒ RR
kk ⇒⇒ KK
Introduction to FEM forIntroduction to FEM for
Deformation AnalysisDeformation Analysis
Computational Geotechnics Finite Element Method in Geotechnical Engineering
6. Discretization of a ContinuumDiscretization of a Continuum
2D modeling:2D modeling:
Computational Geotechnics Finite Element Method in Geotechnical Engineering
7. 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
8. ElementsElements
Different types of 2D elements:Different types of 2D elements:
Computational Geotechnics Finite Element Method in Geotechnical Engineering
9. ElementsElements
Other way of writing:Other way of writing:
uuxx = N= N11 uux1x1 + N+ N22 uux2x2 + N+ N33 uux3x3 + N+ N44 uux4x4 + N+ N55 uux5x5 + N+ N66 uux6x6
uuyy = N= N11 uuy1y1 + N+ N22 uuy2y2 + N+ N33 uuy3y3 + N+ N44 uuy4y4 + N+ N55 uyuy55 + N+ N66 uuy6y6
oror
uuxx == NN uuxx and uand 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
10. StrainsStrains
Strains are the derivatives of displacements. In finite elements they areStrains are the derivatives of displacements. In finite elements they are
determined from the derivatives of the interpolation functions:determined 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 )
εxx =
∂ux
∂x
=a1 +2a3x+a4y=
∂N
∂x
ux
εyy =
∂uy
∂y
=b2 +2b4x+b5y=
∂N
∂y
uy
γxy =
∂ux
∂y
+
∂uy
∂x
=(b1 +a2)+(a4 +2b3)x+(2a5 +b4)y=
∂N
∂x
ux +
∂N
∂y
uy
ε=Bd
Computational Geotechnics Finite Element Method in Geotechnical Engineering
11. Stresses, Constitutive RelationsStresses, Constitutive Relations
Cartesian stress tensor, usuallyCartesian stress tensor, usually
composed in a vector:composed in a vector:
Stresses,Stresses, σσ, are related to strains, are related to strains εε::
σσ == CCεε
In fact, the above relationship is usedIn fact, the above relationship is used
in incremental form:in incremental form:
C is material stiffness matrix andC is material stiffness matrix and
determining material behaviordetermining material behavior
Computational Geotechnics Finite Element Method in Geotechnical Engineering
12. Hooke’s LawHooke’s Law
For simple linear elastic behavior C is based onFor simple linear elastic behavior C is based on
Hooke’s law: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 1
2 −ν 0 0
0 0 0 0 1
2
−ν 0
0 0 0 0 0 1
2 −ν
Computational Geotechnics Finite Element Method in Geotechnical Engineering
13. Hooke’s LawHooke’s Law
Basic parameters in Hooke’s law:Basic parameters in Hooke’s law:
Young’s modulusYoung’s modulus EE
Poisson’s ratioPoisson’s ratio νν
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+ν)
K=
E
3(1−2ν)
Eoed =
E(1−ν)
(1−2ν)(1+ν)
Computational Geotechnics Finite Element Method in Geotechnical Engineering
14. 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 =
∂σ1
∂σ2
ν =−
∂ε3
∂ε1
Eoed =
∂σ1
∂ε1
Computational Geotechnics Finite Element Method in Geotechnical Engineering
axial compression 1D compression
15. Hooke’s LawHooke’s Law
Meaning of parametersMeaning of parameters
in volumetric compressionin volumetric compression
in shearingin shearing
note:note:
K =
∂p
∂εv
G=
∂σxy
∂γxy
σxy≡τxy
Computational Geotechnics Finite Element Method in Geotechnical Engineering
18. Formulation of Stiffness MatrixFormulation of Stiffness Matrix
Formation of element stiffness matrixFormation 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)
coefficientscoefficients αα and position of sample points can be chosen such that the integration is exactand 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= αipi
i=1
n
∑∫
Computational Geotechnics Finite Element Method in Geotechnical Engineering
20. 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:
((ii = step number)= step number)
K∆D=∆R
∆D= K−1
∆R
D= ∆D
i=1
n
∑
21. Solution of EquationsSolution of Equations
From solution of displacementFrom solution of displacement
Strains:Strains:
Stresses:Stresses:
∆D⇒∆d
→∆εi=B∆ui
→σi=σi−1+C∆d
Computational Geotechnics Finite Element Method in Geotechnical Engineering
