Beam, Shell and Plates

8/2/2019 Beam, Shell and Plates

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Contents:

Textbook:References:

Topic 19Beam, Plate, andShell Elements-Part 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 f ive 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, 6.3.5The (degenerate) isoparametric shell and beam elements, including thetransition elements, are presented and evaluated inBathe, 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.


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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, 673-681, 1981.


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STRUCTURAL ELEMENTS Beams Plates ShellsWe 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 degreesof freedom

Topic Nineteen 19-3

Transparency19-1

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Transparency19-3 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 stiffnessI

~ / degree of freedom with\/_ artificial stiffness~ ' / . . zL 5 } ~ - -

xlX1

Stiffness matrix inlocal coordinatesystem (Xi).

Example:Transparency19-4


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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 beam

and 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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Transparency19-7

Transparency19-8

We recall, for the T.L. formulation,f HAtS .. ~ H A t E .. 0dV _ HAt(jJ}Jov 0 II'U 0 II' - ;:ILLinearization

'v OC iirs oe rs 8 0 e i ~ dV +f,v J S i ~ 8o"ly.dV= HAtm - f,v J S i ~ 8 o e y . dV

Also, for the U.L. formulation,J V H A ~ S i j . 8 H A ~ E i j . tdV = HAtm

Linearization

Jv tCifs te rs 8tei} tdV +Jv~ i ~ 8 t ' T \ i j . tdV= H At9R - f",t'Tij. 8tei}tdV


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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

Transparency19-9

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Transparency19-11 time 0

not 90 in generaltime t

Transparency19-122) 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).


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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.

Transparency19-13

Transparency19-14

rX2t v = director vector at node kak = shell thickness at node k(measured into direction of t v ~ )

Shell element geometryE x a m ~ : 9-node element


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Transparency19-15

Element geometry definition: Input mid-surface nodal pointcoordinates. Input all nodal director vectors at time O. Input thicknesses at nodes.

Transparency19-16

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


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Topic Nineteen 19-11

Interpolation of geometry at time 0: Transparency19-17Nk ~ hk Xr ,+ k ~ a hk V ~ i

mid-surface effect of shellonly 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-coordinatet h t k


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Transparency19-19 To obtain the displacements of anymaterial particle,t t 0Ui = Xi - Xi

HenceN NtUi = k ~ hk t u + k ~ ak hk C V ~ i - V ~ i )

wheret u - t x - x I - I I (disp. of nodal point k)

Transparency19-20

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

The incremental displacements fromtime t to time t+a t are, similarly, forany material particle in the shellelement,U. - t+LltX - tx1 - 1 I

whereu = incremental nodal point displacementsV ~ = t + L l t V ~ i - t V ~ i = incremental changein direction cosinesof director vectorfrom time t to timet +Llt


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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 ~ in terms ofnodal point rotations.

We define at each nodal point k thevectors O V and v ~ :- - v r ~ v

vk0\ Ik e2 x n 0Vk2 = 0Vkn X 0Vk1y 1= IIe2 x V ~ 1 I 2 , - - -The vectors v ~ , V and V aretherefore mutually perpendicular.


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Transparency19-23

Transparency19-24

Then let


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Once the incremental nodal pointdisplacements and rotations have beencalculated from the solution of the finiteelement system equilibrium equations,we calculate the new director vectorsusingt + . : l t v ~ = t v +1 ( _ T V ~ dak+T V d ~ k )L

a k , ~ k

and normalize length

Nodal point degrees of freedom: We have only five degrees offreedom per node:- three translations in the Cartesiancoordinate directions- two rotations referred to the localnodal point vectors t v ~ , t v

The nodal point vectors t v ~ , t v change directions in a geometricallynonlinear solution.

Topic Nineteen 1915

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1916 Beam, Plate andShell Elements - Part I

Transparency19-27~

- Node k is sharedby four shell elements

Transparency19-28no physicalstiffness

- Node k is sharedby four shell elements- One director vector

l ~ at node k- No physical stiffnesscorresponding torotation about l ~ ~ .


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~

~

- Node k is sharedby four shell elements- One director vector

t ~ at node k- No physical stiffnesscorresponding torotation about t ~ ~ .


Transparency19-29

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 (threerotational degrees of freedom) or

- impose a boundary rotation (otherthan ak or r3k) at that node.

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Transparency19-31 The above interpolations of Xi, tXi , Uiare employed to establish the straindisplacement transformation matricescorresponding to the Cartesian straincomponents, as in the analysis of 3-Dsolids.

Transparency19-32

Using the expression oe.. derived earlierthe exact linear s t r a i n - d i ~ p l a c e m e n tmatrix JB l is obtained.However, using OUk,i OU k,} to develop thenonlinear strain-displacement matrixJB Nl , 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.


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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."

~ ~

_ es x eter = lies x etl12 ' es = et x erWe note: er, es, et are not mutuallyperpendicular in general.er , es , et are constructed tobe mutually perpendicular.


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Transparency19-35Then the stress-strain law used is, fora linear elastic material,

1 v 01 0osymmetric

000000000

1 - v 0 0-2- kC2v) 0

kC2v)

Transparency19-36

k = shear correction factor

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

QSh = :

usingf 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


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The columns and rows 1 to 3 in CShreflect 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


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Transparency19-39

a) Shell intersectionTransparency19-40

b) Solid-shell intersection


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Resource: Finite Element Procedures for Solids and StructuresKlaus-Jrgen Bathe

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Beam, Shell and Plates