Contents:

Textbook:

References:

Topic 19

Beam, Plate, andShell Elements­Part I

• Brief review of major formulation approaches

• The degeneration of a three-dimensional continuum tobeam and shell behavior

• Basic kinematic and static assumptions used

• Formulation of isoparametric (degenerate) general shellelements of variable thickness for large displacementsand rotations

• Geometry and displacement interpolations

• The nodal director vectors

• Use of five or six nodal point degrees of freedom,theoretical considerations and practical use

• The stress-strain law in shell analysis, transformationsused at shell element integration points

• Shell transition elements, modeling of transition zonesbetween solids and shells, shell intersections

Sections 6.3.4, 6.3.5

The (degenerate) isoparametric shell and beam elements, including thetransition elements, are presented and evaluated in

Bathe, K. J., and S. Bolourchi, "A Geometric and Material NonlinearPlate and Shell Element," Computers & Structures, 11, 23-48, 1980.

Bathe, K. J., and L. W. Ho, "Some Results in the Analysis of Thin ShellStructures," in Nonlinear Finite Element Analysis in StructuralMechanics, (Wunderlich, W., et al., eds.), Springer-Verlag, 1981.

Bathe, K. J., E. Dvorkin, and L. W. Ho, "Our Discrete Kirchhoff and Iso­parametric Shell Elements for Nonlinear Analysis-An Assessment,"Computers & Structures, 16, 89-98, 1983.

•

19-2 Beam, Plate and Shell Elements - Part I

References:(continued)

The triangular flat plate/shell element is presented and also studied in

Bathe, K. J., and L. W. Ho, "A Simple and Effective Element for Anal­ysis of General Shell Structures," Computers & Structures, 13, 673­681, 1981.

STRUCTURAL ELEMENTS

• Beams

• Plates

• Shells

We note that in geometrically nonlinearanalysis, a plate (initially "flat shell")develops shell action, and is analyzedas a shell.

Various solution approaches have been proposed:

• Use of general beam and shelltheories that include the desirednonlinearities.

- With the governing differentialequations known, variationalformulations can be derived anddiscretized using finite elementprocedures.

- Elegant approach, but difficultiesarise in finite element formulations:• Lack of generality• Large number of nodal degrees

of freedom

Topic Nineteen 19-3

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19-4 Beam, Plate and Shell Elements - Part I

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• Use of simple elements, but a largenumber of elements can modelcomplex beam and shell structures.

- An example is the use of 3-nodetriangular flat plate/membraneelements to model complex shells.

- Coupling between membrane andbending action is only introducedat the element nodes.

- Membrane action is not very wellmodeled.

bendingf membraneartificial

.ISs stiffness

I

\~3 / degree of freedom with\/_~ artificial stiffness~'/..z

L5}~--xl

X1

Stiffness matrix inlocal coordinatesystem (Xi).

Example:Transparency

19-4

• Isoparametric (degenerate) beam andshell elements.

- These are derived from the 3-Dcontinuum mechanics equationsthat we discussed earlier, but thebasic assumptions of beam andshell behavior are imposed.

- The resulting elements can beused to model quite general beamand shell structures.

We will discuss this approach in somedetail.

Basic approach:

• Use the total and updated Lagrangianformulations developed earlier.

Topic Nineteen 19-5

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19-6 Beam, Plate and Shell Elements - Part I

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We recall, for the T.L. formulation,

f HAtS.. ~HAtE .. 0dV _ HAt(jJ}Jov 0 II'U 0 II' - ;:IL

Linearization ~

£'v OCiirs oers 80ei~ °dV +f,v JSi~8o"ly. °dV

= HAtm - f,v JSi~8oey.°dV

Also, for the U.L. formulation,

JVHA~Sij.8HA~Eij. tdV = HAtm

Linearization ~

Jv tCifs ters 8tei} tdV +Jv~i~8t'T\ij. tdV

= H At9R - f",t'Tij. 8tei}tdV

• Impose on these equations the basicassumptions of beam and shell -­action:

1) Material particles originally on astraight line normal to the mid­surface of the beam (or shell)remain on that straight linethroughout the response history.

For beams, "plane sections initiallynormal to the mid-surface remainplane sections during the responsehistory".

The effect of transverse sheardeformations is included, andhence the lines initially normal tothe mid-surface do not remainnormal to the mid-surface duringthe deformations.

Topic Nineteen 19-7

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19-8 Beam, Plate and Shell Elements - Part I

Transparency19-11 time 0

not 90° in general

time t

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2) The stress in the direction "normal"to the beam (or shell) mid-surface iszero throughout the response history.

Note that here the stress along thematerial fiber that is initially normalto the mid-surface is considered;because of shear deformations, thismaterial fiber does not remainexactly normal to the mid-surface.

3) The thickness of the beam (or shell)remains constant (we assume smallstrain conditions but allow for largedisplacements and rotations).

Topic Nineteen 19-9

FORMULATION OFISOPARAMETRIC

(DEGENERATE) SHELLELEMENTS

• To incorporate the geometricassumptions of "straight lines normal tothe mid-surface remain straight", and of"the shell thickness remains constant"we use the appropriate geometric anddisplacement interpolations.

• To incorporate the condition of "zerostress normal to the mid-surface" weuse the appropriate stress-strain law.

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rX2

tv~ = director vector at node k

ak = shell thickness at node k(measured into direction of tv~)

Shell element geometryExam~: 9-node element

19-10 Beam, Plate and Shell Elements - Part I

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Element geometry definition:

• Input mid-surface nodal pointcoordinates.

• Input all nodal director vectors at time O.• Input thicknesses at nodes.

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r---X2

- material particle(OXi)

• Isoparametric coordinate system(r, s, t):

- The coordinates rand s aremeasured in the mid-surfacedefined by the nodal pointcoordinates (as for a curvedmembrane element).

- The coordinate t is measured inthe direction of the director vectorat every point in the shell.

s

Topic Nineteen 19-11

Interpolation of geometry at time 0:Transparency

19-17N N

k~' hk °Xr ,+ ~ k~' a~ hk °V~imid-surface effect of shell

only thicknessmaterialparticlewith isoparametriccoordinates (r, s, t)

hk = 2-D interpolation functions (asfor 2-D plane stress, planestrain and axisymmetric elements)

°X~ = nodal point coordinates°V~i = components of °V~

Similarly, at time t, 0t-coordinate

t ~ h t k <D ~ h tvkXi = L.J k Xi + 2 L.J ak k ni~ k=1 vvv k=1 \N\IV

~ I I

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The nodal point coordinates and directorvectors have changed.

X3 0vk_n

}-----+-------/-'--- X2

19-12 Beam, Plate and Shell Elements - Part I

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To obtain the displacements of anymaterial particle,

t t 0Ui = Xi - Xi

HenceN N

tUi = k~1 hk tu~ + ~ k~1 ak hk CV~i - °V~i)

wheretu~ - tx~ - °x~

I - I I (disp. of nodal point k)

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tV~i - °V~i = change in direction cosinesof director vector at node k

The incremental displacements fromtime t to time t+at are, similarly, forany material particle in the shellelement,

U. - t+LltX· - tx·1- 1 I

whereu~ = incremental nodal point displacements

V~i = t+LltV~i - tV~i = incremental changein direction cosinesof director vectorfrom time t to timet +Llt

e3J---X2

To develop the strain-displacementtransformation matrices for the T.L. andU.L. formulations, we need

- the coordinate interpolations for thematerial particles (OXil tXj).

- the interpolation of incrementaldisplacements from the incrementalnodal point displacements androtations.

Hence, express the V~i in terms ofnodal point rotations.

We define at each nodal point k thevectors OV~ and °v~:- - °v~

r~~°v~

°vk0\ Ik e2 x n 0Vk2 = 0Vk

nX 0Vk1

y 1= IIe2 x °V~1I2 ,- - -

The vectors °v~, °V~ and °V~ aretherefore mutually perpendicular.

Topic Nineteen 19-13

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19-14 Beam, Plate and Shell Elements - Part I

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Then let <Xk and ~k be the rotationsabout tv~ and tv~. We have, for small<Xk, ~k,

v~ = - tv~ <Xk + tv~ ~k

>---+--__ t~

t+dtVk_n with elk =0

Hence, the incremental displacementsof any material point in the shellelement are given in terms ofincremental nodal point displacementsand rotations

N N

Ui = L hk u~ + 2t L ak hk [-tV~i <Xk + tV~i ~k]

k=1 k=1

Once the incremental nodal pointdisplacements and rotations have beencalculated from the solution of the finiteelement system equilibrium equations,we calculate the new director vectorsusing

t+.:ltv~ = tv~ +1 (_TV~ dak +TV~ d~k)

Lak,~k

and normalize length

Nodal point degrees of freedom:

• We have only five degrees offreedom per node:- three translations in the Cartesian

coordinate directions- two rotations referred to the local

nodal point vectors tv~, tv~

• The nodal point vectors tv~, tv~

change directions in a geometricallynonlinear solution.

Topic Nineteen 19·15

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19·16 Beam, Plate and Shell Elements - Part I

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~3

- Node k is sharedby four shell elements

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no physicalstiffness

- Node k is sharedby four shell elements

- One director vectorl~~ at node k

- No physical stiffnesscorresponding torotation about l~~.

~1

~3

- Node k is sharedby four shell elements

- One director vectort~~ at node k

- No physical stiffnesscorresponding torotation about t~~.

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• If only shell elements connect tonode k, and the node is notsubjected to boundary prescribedrotations, we only assign fivelocal degrees of freedom to thatnode.

• We transform the two nodal rotationsto the three Cartesian axes in orderto- connect a beam element (three

rotational degrees of freedom) or- impose a boundary rotation (other

than ak or r3k) at that node.

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19-18 Beam, Plate and Shell Elements - Part I

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• The above interpolations of °Xi, tXi , Ui

are employed to establish the strain­displacement transformation matricescorresponding to the Cartesian straincomponents, as in the analysis of 3-Dsolids.

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• Using the expression oe.. derived earlierthe exact linear strain-di~placementmatrix JBl is obtained.

However, using ~ OUk,i OUk,} to develop thenonlinear strain-displacement matrixJBNl, only an approximation to the exactsecond-order strain-displacement rotationexpression is obtained because the inter­nal element displacements depend non­linearly on the nodal point rotations.

The same conclusion holds for the U.L.formulation.

• We still need to impose the conditionthat the stress in the direction"normal" to the shell mid-surface iszero.

We use the direction of the directorvector as the "normal direction."

~s ~s

_ es x eter = lies x etl12 ' es = et x er

We note: er, es, et are not mutuallyperpendicular in general.

er , es , et are constructed tobe mutually perpendicular.

Topic Nineteen 19-19

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19-20 Beam, Plate and Shell Elements - Part I

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Then the stress-strain law used is, fora linear elastic material,

1 v 01 0

o

symmetric

000000000

1-v 0 0-2- k C2v) 0

k C2v)

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k = shear correction factor

whererow 1 (e,)2 (m,)2 (n,)2 elm, m,n, n,e,------------------ --------- ---- ------------ -- ------ -- -----------

QSh = :

using

f 1 = cos (~1, gr) rn1 = cos (~, gr) n1 = cos (~a, gr)f 2 = cos (~1, gs) rn2 = cos (~, gs) n2 = cos (~a, ~)

fa = cos (~1, ~t) rna = cos (~, ~t) na = cos (~a, ~t)

• The columns and rows 1 to 3 in CSh

reflect that the stress "normal" to theshell mid-surface is zero.

• The stress-strain matrix for plasticityand creep solutions is similarlyobtained by calculating the stress­strain matrix as in the analysis of 3-Dsolids, and then imposing thecondition that the stress "normal" tothe mid-surface is zero.

• Regarding the kinematic description ofthe shell element, transition elementscan also be developed.

• Transition elements are elements withsome mid-surface nodes (andassociated director vectors and fivedegrees of freedom per node) andsome top and bottom surface nodes(with three translational degrees offreedom per node). These elementsare used

- to model shell-to-solid transitions- to model shell intersections

Topic Nineteen 19-21

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19-22 Beam, Plate and Shell Elements - Part I

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a) Shell intersectionTransparency

19-40

•

b) Solid-shell intersection

• •

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Resource: Finite Element Procedures for Solids and Structures Klaus-Jürgen Bathe

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