Fe Formulations Hand Out

download Fe Formulations Hand Out

of 24

  • date post

    13-Jul-2016
  • Category

    Documents

  • view

    14
  • download

    6

Embed Size (px)

description

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,