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FE Formulation handout

### Transcript of Fe Formulations Hand Out

• OverviewFinite Element familiesContinuum ElementsStructural Elements

Examples

Finite element formulations

Joel Cugnoni, LMAF / EPFL

March 13, 2013

Joel Cugnoni, LMAF / EPFL Finite element formulations

• OverviewFinite Element familiesContinuum ElementsStructural Elements

Examples

1 Finite Element familiesIntroductionOverview of FE families

2 Continuum Elements3D continuum2D plane strain / stress2D axisymmetry

3 Structural ElementsShell elementsBeam elements

Joel Cugnoni, LMAF / EPFL Finite element formulations

• OverviewFinite Element familiesContinuum ElementsStructural Elements

Examples

IntroductionOverview of FE families

Class of FE formulations

Existing classes of Finite Elements can be characterized by thefollowing criteria:

Geometry modeling:Modeling space: number of coordinates to describe geometry(3D, 2D, 1D)Basic Topology: basic type of topology (solid, surface, wire)

Physical modeling:Physics: the behaviour that is modelled type of DOFs,elementary matrices & resultsPhysical modeling space: 3D, 2D planar / axisymm., 1D number & meaning of DOFs

FE formulation:Element shape: hex, tetra, triangle, wedge, quad.Interpolation: FE shape functions order Number of nodesIntegration: integration scheme (type / order)

Joel Cugnoni, LMAF / EPFL Finite element formulations

• OverviewFinite Element familiesContinuum ElementsStructural Elements

Examples

IntroductionOverview of FE families

Main families of Finite Elements

Joel Cugnoni, LMAF / EPFL Finite element formulations

• OverviewFinite Element familiesContinuum ElementsStructural Elements

Examples

3D continuum2D plane strain / stress2D axisymmetry

3D continuum model

3D continuum model:3D geometry / 3D continuum behaviour / 3D loads, may have

symmetries !!Joel Cugnoni, LMAF / EPFL Finite element formulations

• OverviewFinite Element familiesContinuum ElementsStructural Elements

Examples

3D continuum2D plane strain / stress2D axisymmetry

3D continuum elements

Nodal Coordinate:

xj = {x1, x2, x3}

Nodal DOF:

qj = {u1, u2, u3}

Coordinate Transform:

eT : x = x() = aH() ex

Displacement Interpolation:

euh = eH eq

Joel Cugnoni, LMAF / EPFL Finite element formulations

• OverviewFinite Element familiesContinuum ElementsStructural Elements

Examples

3D continuum2D plane strain / stress2D axisymmetry

Example: linear hexahedron finite element

Master element geometry:

supported by 8 corner nodes, coordinates: (1, 2, 3) [1, 1]3

Basis functions:

h1(1, 2, 3) =18(1 1)(1 2)(1 3)

h2(1, 2, 3) =18(1 + 1)(1 2)(1 3)

h3(1, 2, 3) =18(1 + 1)(1 + 2)(1 3)

h4(1, 2, 3) =18(1 1)(1 + 2)(1 3)

h5(1, 2, 3) =18(1 1)(1 2)(1 + 3)

h6(1, 2, 3) =18(1 + 1)(1 2)(1 + 3)

h7(1, 2, 3) =18(1 + 1)(1 + 2)(1 + 3)

h8(1, 2, 3) =18(1 1)(1 + 2)(1 + 3)

Joel Cugnoni, LMAF / EPFL Finite element formulations

• OverviewFinite Element familiesContinuum ElementsStructural Elements

Examples

3D continuum2D plane strain / stress2D axisymmetry

2D plane strain model

Plane Strain: constrained in longitudinal direction2D geometry / 2D continuum behaviour / 2D loads, infinite

depthJoel Cugnoni, LMAF / EPFL Finite element formulations

• OverviewFinite Element familiesContinuum ElementsStructural Elements

Examples

3D continuum2D plane strain / stress2D axisymmetry

2D plane strain

2D plane strain: definition

The base hypothesis of 2D plane strain problem is:

u1,2 = u1,2(x1, x2) & u3 = 0 33 = 23 = 13 = 0

The constitutive relationship is then: 112212

= E(1 + )(1 2)

1 0 1 00 0 122

112212

Note that 13 = 23 = 0 but 33 6= 0 (Poisson effect):

33 =E(11 + 22)

(1 + )(1 2)Joel Cugnoni, LMAF / EPFL Finite element formulations

• OverviewFinite Element familiesContinuum ElementsStructural Elements

Examples

3D continuum2D plane strain / stress2D axisymmetry

2D plane strain elements

Nodal Coordinate:

xj = {x1, x2}

Nodal DOF:

qj = {u1, u2}

Geometry & displacement interp.:

x = aH(1,2)ex ; euh = eH eq

Stress / strain results:

= {11, 22, 12}T = {11, 22, 12}T

Joel Cugnoni, LMAF / EPFL Finite element formulations

• OverviewFinite Element familiesContinuum ElementsStructural Elements

Examples

3D continuum2D plane strain / stress2D axisymmetry

2D plane stress model

Plane Stress: no constraints in longitudinal direction2D geometry / 2D continuum behaviour / negligible depth

Joel Cugnoni, LMAF / EPFL Finite element formulations

• OverviewFinite Element familiesContinuum ElementsStructural Elements

Examples

3D continuum2D plane strain / stress2D axisymmetry

2D plane stress

2D plane stress: definition

The base hypothesis of 2D plane stress problem is:

33 = 23 = 13 = 0 & u1,2 = u1,2(x1, x2)

The constitutive relationship is then written: 112212

= E(1 2)

1 0 1 00 0 12

112212

Note that 33 6= 0 and is derived from the Poisson effects:

33 = E

(11 + 22)

Joel Cugnoni, LMAF / EPFL Finite element formulations

• OverviewFinite Element familiesContinuum ElementsStructural Elements

Examples

3D continuum2D plane strain / stress2D axisymmetry

2D plane stress elements

Nodal Coordinate:

xj = {x1, x2}

Nodal DOF:

qj = {u1, u2}

Geometry & displacement interp.:

x = aH(1,2)ex ; euh = eH eq

Stress / strain results:

= {11, 22, 12}T = {11, 22, 12}T

Joel Cugnoni, LMAF / EPFL Finite element formulations

• OverviewFinite Element familiesContinuum ElementsStructural Elements

Examples

3D continuum2D plane strain / stress2D axisymmetry

2D axisymmetric model

Axisymmetric model: revolution geometry2D axisymm. geometry / 3D continuum behaviour / 2D axisymm.

loadsJoel Cugnoni, LMAF / EPFL Finite element formulations

• OverviewFinite Element familiesContinuum ElementsStructural Elements

Examples

3D continuum2D plane strain / stress2D axisymmetry

2D axisymmetric elasticity

2D axisymmetric elasticity

2D axisymmetric models are written in cylindrical coordinates{x1,2,3} {r , z , }. The axisymmetric problem derives from thehypotheses that it is invariant with coordinate and thus thedisplacement, stress & strains fields depend only on thecoordinates r and z .

ur = ur (r , z); uz = uz(r , z); u = 0

rr =urr

; zz =uzz

; =urr

; rz =urz

+uzr

; r = z = 0

rr , zz , rz , = f (r , z); r = z = 0

Note that, even if we have reduced the dimensionnality of theproblem, the constitutive behaviour is fully 3D.

Joel Cugnoni, LMAF / EPFL Finite element formulations

• OverviewFinite Element familiesContinuum ElementsStructural Elements

Examples

3D continuum2D plane strain / stress2D axisymmetry

2D axisymmetric elements

Nodal Coordinate:

xj = {x1, x2} = {r , z}

Axis of symmetry = OX2 Nodal DOF:

qj = {u1, u2} = {ur , uz}

Geometry & displacement interp.:

x = aH(1,2)ex ; euh = eH eq

Stress / strain results:

= {rr , zz , , rz}T = {rr , zz , , rz}T

Joel Cugnoni, LMAF / EPFL Finite element formulations

• OverviewFinite Element familiesContinuum ElementsStructural Elements

Examples

Shell elementsBeam elements

Shell elements

Shell part (3D geometry / 2.5D structural behaviour)

Joel Cugnoni, LMAF / EPFL Finite element formulations

• OverviewFinite Element familiesContinuum ElementsStructural Elements

Examples

Shell elementsBeam elements

Shell elements

Nodal Coordinates, thickness, normal vector :

xj = {x1, x2, x3} ; nj ; t jGeometric interpolation:

x() =i

ahi (1, 2)(exi +

1

23 t

i ni)

Nodal DOF:qj = {u1, u2, u3, ur1, ur2(, ur3)}

Displacement interpolation:

euh() =ai

hi (1, 2)(e ui +

1

23 t

i[ur1 v1 + ur2 v2)

]Joel Cugnoni, LMAF / EPFL Finite element formulations

• OverviewFinite Element familiesContinuum ElementsStructural Elements

Examples

Shell elementsBeam elements

Beam elements

Wire part (3D geometry / 1.5D structural behaviour)

Joel Cugnoni, LMAF / EPFL Finite element formulations

• OverviewFinite Element familiesContinuum ElementsStructural Elements

Examples

Shell elementsBeam elements

Beam elements

Nodal Coordinates, dimensions, normal vectors :

xj = {x1, x2, x3} ; nj2; nj3; t j2; t j3Geometric interpolation:

x() =i

ahi (1)(exi +

1

22 t

i2 n

i2 +

1

23 t

i3 n

i3)

Nodal DOF:qj = {u1, u2, u3, ur1, ur2, ur3)}

(rotations / displacement expressed in the global coord. system)

Joel Cugnoni, LMAF / EPFL Finite element formulations

• OverviewFinite Element familiesContinuum ElementsStructural Elements

Examples

Example 1: TGV Bogie

Suitable modeling methods:

3D solids, use directly the 3D CAD model

3D shells (relatively thin plates), need to build surface model

Joel Cugnoni, LMAF / EPFL Finite element formulations

• OverviewFinite Element familiesContinuum ElementsStructural Elements

Examples

Example 2: Aircraft fuselage

Suitable modeling methods:

3D shells (thin skins), need to build surface model3D solids,