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


Examples

1 Finite Element familiesIntroductionOverview of FE families

2 Continuum Elements3D continuum2D plane strain / stress2D axisymmetry

3 Structural ElementsShell elementsBeam 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)



Examples

IntroductionOverview of FE families

Main families of Finite 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


Examples


3D continuum elements

Nodal Coordinate:

xj = {x1, x2, x3}

Nodal DOF:

qj = {u1, u2, u3}

Coordinate Transform:

eT : x = x(ξ) = aH(ξ) ex

Displacement Interpolation:

euh = eH eq



Examples


Example: linear hexahedron finite element

Master element geometry:

supported by 8 corner nodes, coordinates: (ξ1, ξ2, ξ3) ∈ [−1, 1]3

Basis functions:

h1(ξ1, ξ2, ξ3) = 18(1− ξ1)(1− ξ2)(1− ξ3)

h2(ξ1, ξ2, ξ3) = 18(1 + ξ1)(1− ξ2)(1− ξ3)

h3(ξ1, ξ2, ξ3) = 18(1 + ξ1)(1 + ξ2)(1− ξ3)

h4(ξ1, ξ2, ξ3) = 18(1− ξ1)(1 + ξ2)(1− ξ3)

h5(ξ1, ξ2, ξ3) = 18(1− ξ1)(1− ξ2)(1 + ξ3)

h6(ξ1, ξ2, ξ3) = 18(1 + ξ1)(1− ξ2)(1 + ξ3)

h7(ξ1, ξ2, ξ3) = 18(1 + ξ1)(1 + ξ2)(1 + ξ3)

h8(ξ1, ξ2, ξ3) = 18(1− ξ1)(1 + ξ2)(1 + ξ3)



Examples


2D plane strain model

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

depthJoel Cugnoni, LMAF / EPFL Finite element formulations


Examples


2D plane strain

2D plane strain: definition

The base hypothesis of 2D plane strain problem is:

u1,2 = u1,2(x1, x2) & u3 = 0 ⇔ ε33 = ε23 = ε13 = 0

The constitutive relationship is then: σ11σ22σ12

=E

(1 + ν)(1− 2ν)

1− ν ν 0ν 1− ν 00 0 1−2ν

2

ε11ε22ε12

Note that σ13 = σ23 = 0 but σ33 6= 0 (Poisson effect):

σ33 =Eν(ε11 + ε22)

(1 + ν)(1− 2ν)



Examples


2D plane strain elements

Nodal Coordinate:

xj = {x1, x2}

Nodal DOF:

qj = {u1, u2}

Geometry & displacement interp.:

x = aH(ξ1,2) ex ; euh = eH eq

Stress / strain results:

σ = {σ11, σ22, σ12}T ε = {ε11, ε22, ε12}T



Examples


2D plane stress model

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



Examples


2D plane stress

2D plane stress: definition

The base hypothesis of 2D plane stress problem is:

σ33 = σ23 = σ13 = 0 & u1,2 = u1,2(x1, x2)

The constitutive relationship is then written: σ11σ22σ12

=E

(1− ν2)

1 ν 0ν 1 00 0 1−ν

2

ε11ε22ε12

Note that ε33 6= 0 and is derived from the Poisson effects:

ε33 = − νE

(σ11 + σ22)



Examples


2D plane stress elements

Nodal Coordinate:

xj = {x1, x2}

Nodal DOF:

qj = {u1, u2}




σ = {σ11, σ22, σ12}T ε = {ε11, ε22, ε12}T



Examples


2D axisymmetric model

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

loadsJoel Cugnoni, LMAF / EPFL Finite element formulations


Examples


2D axisymmetric elasticity

2D axisymmetric elasticity

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

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

εrr =∂ur

∂r; εzz =

∂uz

∂z; εθθ =

ur

r; εrz =

∂ur

∂z+∂uz

∂r; εrθ = εzθ = 0

σrr , σzz , σrz , σθθ = f (r , z); σrθ = σzθ = 0

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



Examples


2D axisymmetric elements

Nodal Coordinate:

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

Axis of symmetry = OX2 Nodal DOF:

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




σ = {σrr , σzz , σθθ, σrz}T ε = {εrr , εzz , εθθ, εrz}T



Examples

Shell elementsBeam elements

Shell elements

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



Examples


Shell elements

Nodal Coordinates, thickness, normal vector :

xj = {x1, x2, x3} ; nj ; t j

Geometric interpolation:

x(ξ) =∑i

ahi (ξ1, ξ2)( exi +1

2ξ3 t i ni)

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

Displacement interpolation:

euh(ξ) =a∑i

hi (ξ1, ξ2)( e ui +1

2ξ3 t i

[−ur1 v1 + ur2 v2)

]Joel Cugnoni, LMAF / EPFL Finite element formulations


Examples


Beam elements

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



Examples


Beam elements

Nodal Coordinates, dimensions, normal vectors :

xj = {x1, x2, x3} ; nj2; nj3; t j2; t j3

Geometric interpolation:

x(ξ) =∑i

ahi (ξ1)( exi +1

2ξ2 t i2 ni2 +

1

2ξ3 t i3 ni3)

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

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



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



Examples

Example 2: Aircraft fuselage

Suitable modeling methods:

3D shells (thin skins), need to build surface model3D solids, ok but will require more computation time

Note: if windows are neglected and section is constrained in length direction, 2D plane strain model can be

considered.



Examples

Example 3: Stiffener in a sailing boat

Suitable modeling methods

2D plane stress is the best choice (no constraints in thicknessdirection)3D shells, good choice, necessary for buckling analysis3D solids, ok but more computation cost



Examples

Example 4: Pressure Vessel

Suitable modeling methods

3D solids, no simplification but one may assumequasi-symmetry3D shells, ok but thickness/length ratio is high2D axisymmetric, ok but need to neglect side hole


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