accurately described using incremental Þnite Keywordsrecurdyn.cadmen.com/paper/19064394.pdf · Analysis of thin plate structures using the absolute nodal coordinate formulation K

Analysis of thin plate structures using theabsolute nodal coordinate formulationK Dufva1 and A A Shabana2�

1Department of Mechanical Engineering, Lappeenranta University of Technology, Lappeenranta, Finland2Department of Mechanical Engineering, University of Illinois at Chicago, Chicago, Illinois, USA

The manuscript was received on 10 January 2005 and was accepted after revision for publication on 9 May 2005.

DOI: 10.1243/146441905X50678

Abstract: The absolute nodal coordinate formulation can be used in multibody systemapplications where the rotation and deformation within the finite element are large andwhere there is a need to account for geometrical non-linearities. In this formulation, the gradi-ents of the global positions are used as nodal coordinates and no rotations are interpolated overthe finite element. For thin plate and shell elements, the plane stress conditions can be appliedand only gradients obtained by differentiation with respect to the element mid-surface spatialparameters need to be defined. This automatically reduces the number of element degrees offreedoms, eliminates the high frequencies due to the oscillations of some gradient componentsalong the element thickness, and as a result makes the plate element computationally moreefficient. In this paper, the performance of a thin plate element based on the absolute nodalcoordinate formulation is investigated. The lower dimension plate element used in this investi-gation allows for an arbitrary rigid body displacement and large deformation within theelement. The element leads to a constant mass matrix and zero Coriolis and centrifugalforces. The performance of the element is compared with other plate elements previouslydeveloped using the absolute nodal coordinate formulation. It is shown that the finite elementused in this investigation is much more efficient when compared with previously proposedelements in the case of thin structures. Numerical examples are presented in order to demon-strate the use of the formulation developed in this paper and the computational advantagesgained from using the thin plate element. The thin plate element examined in this study canbe efficiently used in many applications including modelling of paper materials, belt drives,rotor dynamics, and tyres.

Keywords: large deformation, thin plate elements, absolute nodal coordinate formulation,multibody applications

1 INTRODUCTION

The finite element method is often used to solve thedeformation problems in many multibody systemapplications. The large rotations that characterizethe body motion in such systems cannot be

accurately described using incremental finiteelement formulations that employ linearization andinfinitesimal rotations as nodal coordinates. In con-trast, methods based on the large rotation vectorapproach do not lead to a unique rotation field andsuffer from the problem of coordinate redundancyand energy drift. The problems associated with theuse of the incremental procedures and large rotationvector formulations in multibody system appli-cations are discussed in the literature [1, 2]. Whenthe deformations are small, the floating frame of

�Corresponding author: Department of Mechanical Engineering

(M/C 251), University of Illinois at Chicago, 2031 Engineering

Research Facility, 842 West Taylor Street, Chicago, Illinois

60607-7022, USA.

345

JMBD2 # IMechE 2005 Proc. IMechE Vol. 219 Part K: J. Multi-body Dynamics

reference formulation that employs assumed modeshapes is commonly used in multibody system appli-cations. For large deformation problems, the absol-ute nodal coordinate formulation has been used inmany applications [3, 4]. In the absolute nodal coor-dinate formulation, the bodies are discretized usingfinite elements as in the classical finite elementapproach, but gradients of absolute position vectorsinstead of rotations are used as nodal coordinates.Using this coordinate description, one obtains aconstant mass matrix for the finite element andzero Coriolis and centrifugal forces. The stiffnessmatrix, on the contrary, is highly non-linear functionof the element nodal coordinates [5]. Several plateelements based on the absolute nodal coordinateformulation have been proposed in the literaturefor solving large deformation problems [2, 6, 7].The results presented in some of these investigationswere experimentally verified and demonstrated thatthe proposed plate and shell elements can be effec-tively and efficiently used in solving large defor-mation problems in multibody system applicationsin which the components undergo finite rotations.The success of using the proposed plate elementsbased on the absolute nodal coordinate formulationin the analysis of the large rotation and deformationof very flexible bodies can be attributed to manyfactors, which can be summarized as follows. Theabsolute nodal coordinate formulation can beimplemented in the framework of a non-incrementalsolution procedure; the rotation field is uniquelydefined, the mass matrix is constant, and the Coriolisand centrifugal forces are equal to zero, and in thisformulation, there is no restriction on the amountof rotation or deformation within the finite element.

Although the absolute nodal coordinate formu-lation has been successfully used in the analysis ofvery flexible bodies, numerical difficulties are encoun-tered when the multibody system includes very thinand stiff components. It was observed that for thinand stiff structures, the oscillations of some of the gra-dient components along the element thickness intro-duce very high frequencies that make the absolutenodal coordinate formulation less efficient; in someextreme cases, the solution can only be obtainedusing implicit integration methods. The purpose ofthis study is to examine the performance of a compu-tationally efficient reduced-order finite plate elementfor thin and stiff structures [7]. Formulation of theelement elastic forces is based on the classicalapproach where plate bending and plane stress con-ditions are applied. Previous absolute nodal coordi-nate formulation plate and shell elements are basedon the element local coordinate system and planestress assumption or a general continuum mechanicsapproach [2]. When a general continuum mechanicsapproach is applied, no assumptions are made

regarding the cross-section deformation as in thecase when the classical plate theories are used. Thisapproach can be used when the ratio of the elementthickness to its length is high. In the case of thin andstiff structures, numerical problems due to defor-mation of the cross-section can be encountered. Thereduced-order element used in this investigationdoes not suffer from the aforementioned numericalproblems and some computer models based on thisreduced-order element can be more than 100 timesfaster than models that are based on the elementsthat employ full parameterization.

Generalization of the plate elements used in thefinite element formulations to the absolute nodalcoordinate formulation is proposed in reference[7, 8]. Kirchhoff plate theory that does not accountfor the shear deformation is assumed and theelements developed in these investigations are wellsuited for thin plate applications. Different descrip-tions for the element elastic forces in longitudinaland transverse directions have also been proposed.Two-dimensional interpolation functions are usedand owing to the need for second derivatives, 48degrees of freedom were required [7]. In the work ofDmitrochenko and Pogorelov [7], the use of differentset of shape functions to obtain an element with 36degrees of freedom without second derivatives wasdiscussed without providing the exact element formu-lation or investigating numerically its performance. Inthe reduced-order thin plate element examined in thisstudy, two-dimensional shape functions are used toformulate an element with 36 degrees of freedom.The performance of the new element is tested andthe obtained results are used to compare this elementwith previously proposed plate elements based on theabsolute nodal coordinate formulation. It is shownthat the proposed low-order element does not sufferfrom the high oscillation problem which resultsfrom the variation of the gradients used in the pre-viously proposed elements along the element thick-ness. On the basis of the results obtained in thisinvestigation that demonstrates that the thin elementformulation can be more than 100 times faster thanthe elements that employ full parameterization, thisthin element formulation can be efficiently used inother applications including rotor dynamics, papermodelling, belt drives, and tyres.

2 ELEMENT KINEMATICS

In this section, both the reduced order 36-degrees-of-freedom and the 48-degrees-of-freedom plateelements are reviewed in order to explain the basicdifferences between the two elements. The rigidbody modes of the reduced-order element are alsochecked in this section.

346 K Dufva and A A Shabana

Proc. IMechE Vol. 219 Part K: J. Multi-body Dynamics JMBD2 # IMechE 2005

2.1 Complete parameterization

In previous investigations, a higher-order plateelement with 48 degrees of freedom was proposed[2]. This higher-order element, which does not restrictthe modes of deformation of the element cross-section, can be used to relax the assumptions ofclassical plate and shell theories. Figure 1 shows theglobal and local coordinates used to define the absol-ute position and gradient coordinates in the absolutenodal coordinate formulation. The global positionvector r of the material point P on the plate elementcan be defined using the element shape functionsand the nodal coordinate vector as follows

r ¼ S(x, y, z)e (1)

where S is the element shape function matrixexpressed in terms of the element spatial coordinatesx, y, and z and e is the vector of nodal coordinates thatconsist of nodal positions and slope coordinates.These positions and slope coordinates are absolutevariables defined in global inertial frame. In theelement proposed by Mikkola and Shabana [2], theshape functions are functions of the three spatialelement parameters x, y, and z. This representationallows for the deformation of the element cross-section. For a node n, the nodal coordinate vector isdefined as follows

en ¼ rnT @rnT

@x

@rnT

@y

@rnT

@z

� �T

(2)

Using Fig. 1, the total vector of the element nodalcoordinates can be written as follows

e ¼ eAT

eBT

eCT

eDT� �T

(3)

For plate and shell elements, it is not, in general,necessary to ensure the continuity of the coordinatesat the element interface. This problem was addressedin reference [2] by proposing two different shapefunction matrices, SA and SB. The shape functionmatrix SA, which is obtained using incompletefourth-order interpolation polynomials, does notguarantee the continuity of the coordinates at the

element interface. In order to ensure element com-patibility, the shape function matrix SB was proposedand used to define an element that satisfies the con-vergence requirements and is capable of describingthe rigid body motion and the case of constantstrain. The first shape function matrix, SA, on thecontrary, does not meet the requirements for comple-teness and monotonic convergence. The numericalresults obtained using several examples, however,show that the results obtained using the two shapefunction matrices are in good agreement [2]. Bothshape functions lead to an element with 48 degreesof freedom.

2.2 Thin plate

For thin plates, the deformation of the element alongthe thickness direction can be neglected. This leadsto reduced set of deformation modes as the displace-ment field of the element becomes dependent on thespatial coordinates x and y only. In this case, theposition vector gradients obtained by differentiationwith respect to z are not considered as nodal coordi-nates, leading to a reduced-order element with36 degrees of freedom. The normal of the mid-surface of the plate can always be defined usingcross product of the vectors rx and ry, with subscriptsx and y referring to partial derivatives with respectto these coordinates. Shape functions can be directlyobtained from the shape function matrix SA byomitting the components that depend on the z coor-dinate. The obtained shape function matrix thatdepends only on the coordinates x and y is referredto as SC and is presented in Appendix 2. For thereduced-order element, the element nodal coordi-nate vector at node n is defined as follows

en ¼ rnT @rnT

@x

@rnT

@y

� �T

(4)

The reduced element is of the non-conformingtype and the continuity of the gradients at the inter-face between adjoined elements is not ensured. Notethat with the element that employs full parameteri-zation and nine gradient coordinates for each node,continuity of the displacement gradients can beensured at the mid-surface interfaces when usingthe shape function matrix SB.

2.3 Rigid body motion

In the remainder of this section, the capability of thereduced-order element to represent an arbitrary rigidbody motion is demonstrated. To this end, a generalthree-dimensional displacement that can be expressedin terms of a translation of a reference point and threeFig. 1 Plate element dimensions and coordinates

Analysis of thin plate structures 347


rotations ux, uy, and uz about the element x, y, and zaxes, respectively, is considered. For a general three-dimensional rotation, the global position vector ofan arbitrary point on the plate element due torotations only can be written as follows

u ¼ Aub (5)

where ub ¼ x y 0� �T

is the position vector of anarbitrary point on the mid-surface defined in theelement coordinate system. The rotation matrix A interms of the three Euler angles is written as follows

A ¼

cos uz cos uy � sin uz cos ux þ cos uz sin uy sin ux

sin uz cos uy cos uz cos ux þ sin uz sin uy sin ux

� sin uy cos uy sin ux

264

sin uz sin ux þ cos uz sin uy cos ux

� cos uz sin ux þ sin uz sin uy cos ux

cos uy cos ux

375

Columns of the rotation matrix, A, are denoted,respectively, as i, j, and k. From equation 5, it followsthat the global position of point P due to an arbitraryrotational motion can be written as

u ¼

x cos uz cos uy þ y(� sin uz cos ux

þ cos uz sin uy sin ux)

x sin uz cos uy þ y( cos uz cos ux

þ sin uz sin uy sin ux)

� sin uyx þ y cos uy cos ux

2666664

3777775

(6)

The global position of the arbitrary point as the resultof a general rigid body displacement can be written asfollows

r ¼ Rþ Aub (7)

where R is the global position vector of the origin ofthe element coordinate system. The vector of nodalcoordinates at node n as the result of the rigid bodymotion is as follows

en ¼ RnT

iT jT� �T

(8)

where Rn is the global position of the node that can bedefined using equation (7). In the absolute nodal coor-dinate formulation, the location of the arbitrary pointon the plate element using equations (1) and (8) canbe obtained as

r ¼ S(x, y)e

¼

R1 þ x cos uz cos uy þ y(� sin uz cos ux

þ cos uz sin uy sin ux)

R2 þ x sin uz cos uy þ y( cos uz cos ux

þ sin uz sin uy sin ux)

R3 � sin uyx þ y cos uy cos ux

26666664

37777775

(9)

Using equations (1), (6), and (8), it is clear that thereduced-order element can describe exact rigid bodymotion when absolute positions and slopes are usedas nodal coordinates.

3 FORMULATION OF THE ELASTIC FORCES

In this section, the formulations of the elastic forcesfor the high- and reduced-order elements are pre-sented. Special formulation needs to be used in thecase of the reduced-order element because not allthe position vector gradients are present.

3.1 General formulation

The Lagrangian strain tensor can be written as

1m ¼1

2(JTJ� I) (10)

where subscript m is used to indicate the matrix formof the Lagrangian strains and J is the matrix of theposition vector gradients defined as

J ¼@r

@j¼

@r1

@j1

@r1

@j2

@r1

@j3

@r2

@j1

@r2

@j2

@r2

@j3

@r3

@j1

@r3

@j2

@r3

@j3

26666664

37777775

(11)

In this equation

r ¼ r1 r2 r3

� �T, j ¼ j1 j2 j3

� �T¼ Se0 (12)

where e0 is the vector of nodal coordinates in theinitial undeformed configuration. Using equation(10), the strain vector can be defined as

1 ¼ 111 122 133 112 113 123

� �T¼

1

2½(rT

j1rj1� 1) (rT

j2rj2� 1) (rT

j3rj3� 1)

2rTj1

rj2rTj1

rj32rT

j2rj3�T

(13)

If the finite element spatial coordinates aredenoted by the vector x given in terms of itscomponents as

x ¼ x y z� �T

¼ ½x1 x2 x3�T (14)

one can then define the following transformation

@x

@j¼

@j

@x

� ��1

¼ a1 a2 a3

� �(15)



It follows from chain rule of differentiation that

rji¼@r

@ji

¼@r

@xai ¼ rxai (16)

Using this equation, the vector of Lagrangianstrains can be written in terms of gradients definedin the element coordinate system as follows

1 ¼1

2(aT

1 rTx rxa1 � 1) (aT

2 rTx rxa2 � 1) (aT

3 rTx rxa3 � 1)

�

2aT1 rT

x rxa2 2aT1 rT

x rxa3 2aT2 rT

x rxa3

�T(17)

The derivatives of the strain components withrespect to the element coordinates enter in theformulation of the elastic forces and need to beevaluated. Note that

rxai ¼ a1iSx1þ a2iSx2

þ a3iSx3

� �e ¼ Deie (18)

where

Dei ¼ a1iSx1þ a2iSx2

þ a3iSx3

� �(19)

and Sxi¼ @S=@xi. It follows that

@(rxai)

@e¼ Dei (20)

In terms of Dei, the strain vector can be written asfollows

1 ¼1

2½eTDT

e1De1e� 1 eTDTe2De2e� 1 eTDT

e3De3e� 1

2eTDTe1De2e 2eTDT

e1De3e 2eTDTe2De3e�T (21)

Therefore, the derivatives of the strain com-ponents with respect to the vector of nodal coordi-nates can be defined as

@1

@e¼

eTDTe1De1

eTDTe2De2

eTDTe3De3

eT(DTe1De2 þDT

e2De1)

eT(DTe1De3 þDT

e3De1)

eT(DTe2De3 þDT

e3De2)

266666664

377777775

(22)

This matrix is linear in the vector of nodal coordi-nates e, whereas the strain vector is a quadraticfunction of e.

The vector of elastic forces can be obtained usingthe virtual work, which can be written as

dWe ¼

ðV

1TEd1jJjdV (23)

where E is the matrix of elastic coefficients, and V isthe volume. The second Piola–Kirchoff stresstensor is used to obtain this expression. Usingequations (21) and (22), the virtual work of the elasticforces can be written as follows

dWe ¼ QTkde (24)

where the vector of element generalized elasticforces Qk is defined as

Qk ¼

ðV

@1

@e

� �T

E1jJjdV (25)

The strain vector 1 can be written as

1 ¼ 1J þ Im (26)

where

1J ¼1

2½aT

1 rTx rxa1 aT

2 rTx rxa2 aT

3 rTx rxa3

2aT1 rT

x rxa2 2aT1 rT

x rxa3 2aT2 rT

x rxa3�T (27)

and

Im ¼ �1

21 1 1 0 0 0� �T

(28)

Using the definitions in the preceding twoequations and denoting EIm ¼ Em, one can writethe vector of elastic forces in the following form

Qk ¼

ðV

@1

@e

� �T

E1J jJjdV þ

ðV

@1

@e

� �T

EmjJjdV (29)

It is clear that the vector of elastic forces is a non-linear function of the nodal coordinates.

3.2 Thin plate formulation

For the thin plate based on the absolute nodalcoordinate formulation, the vector of the elementelastic forces can be derived using the strain energyfunction. In this study, the plane stress conditionsare assumed for the membrane stiffness, whereasthe bending stiffness of the element is accounted forusing the curvature of the element mid-plane [6, 7].In the general continuum mechanics approach pre-viously presented in this section, the plane stressassumptions are not made and for thin plates, theuse of this general approach leads to high numericalstiffness because of the oscillation of some gradientcomponents along the element thickness. Elementlocal coordinate systems may also be used to definethe element elastic forces [6], but the use of suchlocal frame does not simplify the formulation andthe element stiffness matrix remains highly non-linear. Furthermore, the use of an element localcoordinate system requires the use of non-linear



strain–displacement relation in the case of largedeformation.

Non-linear Green–Lagrange strain measure isemployed in order to account for geometricallynon-linear behaviour and to ensure zero strainunder rigid body motion. Strain components areexpressed using gradients obtained by differentiationwith respect to global spatial coordinates as definedby equation (11). If the element coordinate systemis assumed to be initially parallel to the globalcoordinate system, the transformation matrix ofequation (15) is the identity matrix. In this article,only initially undeformed elements are considered.The Lagrangian strain tensor can be defined usingthe matrix of position vector gradients J as follows

1m ¼1

2(JTJ� I) (30)

where I is the 3�3 identity matrix. For plane stressconditions, the stresses in the plate thicknessdirection are assumed to be zero and the strains inthis direction are expressed as function of the strainsat the element mid-surface. The strain vector at themid-surface can be obtained from equation (30) asfollows

1 ¼ 1m111m22

21m12

� �T¼ 1xx 1yy 21xy

� �T(31)

where 1xx and 1yy are the normal strain componentsin x and y directions, respectively, and 1xy is theshear strain. In order to account for the bending stiff-ness, the correct definition for the element mid-surface curvature needs to be defined. To define thecurvature in terms of the gradient vectors, the follow-ing relations are used [7]

Kxx ¼rT

xxn

knk3, Kyy ¼

rTyyn

knk3, Kxy ¼

rTxyn

knk3(32)

where n is the normal to the element mid-surfaceobtained as n ¼ rx � ry.

If the plate element is initially curved, the strainenergy density function must be integrated withrespect to the undeformed curved reference con-figuration. Relation between volumes in the un-curved reference and initially curved configurationcan be defined using constant transformation andcan be expressed as follows

Vo ¼@x

@j

� �V (33)

where Vo is the volume of the element in the initiallycurved configuration and V is volume in theuncurved configuration. The strain energy of the

element can now be written as follows [9]

U ¼1

2

ðVo

1TE1 dVo þ1

2

ðVo

kTEk dVo (34)

where E is the matrix of elastic coefficients obtainedusing the plane stress conditions [10], and the strainvector is defined using equation (31). The vectorof curvatures multiplied by constant thickness ofthe element defines k ¼ z kxx kyy 2kxy

� �T. In

problems where the membrane stresses aredominant, the effect of the curvature can beneglected. In the absolute nodal coordinate formu-lation, the vector of elastic forces can be obtainedas follows

Qk ¼@U

@e

� �T

(35)

where Qk is the vector of the element elasticforces.

4 EQUATIONS OF MOTION

The use of equation (1) for thin plates leads to a con-stant mass matrix and zero Coriolis and centrifugalforces. Differentiating equation (1) with respect totime, the element kinetic energy is obtained asfollows

T ¼1

2

ðVo

r_rT_r dVo (36)

where _r is the velocity vector of an arbitrary point onthe plate, r the material density, and Vo the elementinitial volume. Equation (36) leads to a constantmass matrix defined as follows

M ¼

ðVo

rSTC SC dVo

� �(37)

The mass matrix remains constant under an arbi-trary large displacement. The equations of motioncan be written in terms of the constant mass matrixM, the non-linear element nodal forces Qk, and theapplied external nodal forces, Qe, as follows [2]

Meþ Qk ¼ Qe (38)

where e is the vector of nodal accelerations. As themass matrix is constant, one can use the Choleskycoordinates to define an identity inertia matrix asdiscussed in previous publications [11].



5 NUMERICAL EXAMPLES

The performance of the low-order element isdemonstrated in this section using numericalexamples. The results obtained are compared withthe results obtained using the element proposed byMikkola and Shabana [2]. The improved accuracyas the result of using the low-order element for thinplates employing the definition of the curvature isfirst demonstrated using a static problem. Thereduced-order element with shape function matrixSC is referred to in this section as Model I and thefull parameterization element with all nine gradientcoordinates is referred to as Model II. A simplecantilever plate structure is subjected to two forcecomponents at the free end as shown in Fig. 2. Thecantilever structure has dimensions of 0.5, 0.15,and 0.001 m, respectively, for the length l, width w,and thickness h. The material is assumed steelwith Young’s modulus of 2.07 � 1011 Pa andPoisson’s ratio of 0.3. The applied force F is 30 N.Tests were made with 1, 2, 4, 8, and 16 elements.Displacements of the free end tip are shown inFig. 3. It is clear from the results presented in thisfigure that the low-order element has betteraccuracy and convergence characteristics becauseof the use of the curvature definition in the thinplate model. Smaller number of elements is requiredfor convergence when compared with the higher-order element.

The performance of the low-order element indynamic analysis is examined using a flexible pendu-lum. Results obtained using the same pendulum arereported in previous publications [2, 7]. Young’smodulus of the material is assumed 1.0 � 105 Pa,material density is 7810 kg/m3, and Poisson’s ratiois 0.3. The pendulum has a length of 0.3 m, a widthof 0.3 m, and a thickness of 0.01 m. The pendulumis simulated for 0.3 s using 1, 22, 42, and 82 elements.Boundary conditions for the pendulum are appliedat the corner node. In order to obtain the constraintsfor the spherical joint, all translation displace-ments of the node are fixed while rotations are free.This is accomplished using constraint equations forthe position coordinates of the node. The equations

of motion are solved using the software Matlab andthe integrator ode23tb. Simulation times for thependulum example with a thickness of 0.01 m arepresented in Table 1. The results are obtained usingan Intel Pentium 4 PC with 2.8 GHz processor.Figure 4 shows the results obtained using the twomodels for the position of the point A of thependulum.

For thin structures, the element with all ninegradients leads to numerical problems. The relationbetween the element thickness and the computationtime is studied using the same pendulum exampleusing the 22 element model. The element thicknessis varied from 0.01 to 0.0005 m. The simulationtimes for the two models are presented in Table 2.It is clear from the results presented in this tablethat the simulation time for the low-order elementdoes not appreciably change as the element thick-ness decreases, whereas the simulation time for thehigher-order element significantly increases as theelement thickness decreases. It is important to notethat as the thickness decreases, the low-orderelement model is more than 100 times faster thanthe higher-order element. Figure 5 shows the

Table 1 Simulation times as a function of the element

numbers

Number ofelements

Ratio ofsimulation time

Simulation time (s)

Model I Model II

1 4.0 2 822 6.0 7 4242 6.6 133 88582 5.2 8340 43 700

Fig. 3 Tip displacement as a function of the element

numbers (†, Model I; V, Model II)

Fig. 2 Thin cantilever plate



position coordinates of points A and B as predictedby the two models for thickness 0.01 m, whereasthe deformed shape of the flexible pendulum withthis thickness is depicted in Fig. 6. Note that, becauseof the symmetry, the X and Y coordinates are thesame in the result presented in Fig. 5. The deformedshapes are depicted using the 1 and 82 elementmodels. The obtained results are also found to bein good agreement with those obtained by Dmitro-chenko and Pogorelov [7].

Bending moments and shear forces can beobtained from the stress distribution by integrationover the element thickness. These moments andforces can be calculated after the finite elementsolution for displacements is obtained. Generally, inthe finite element analysis, the stresses are obtainedaccurately only at integration points and can be

interpolated over the element. Bending momentsand shear forces are then obtained from stress distri-bution or directly at the integration points.

6 SUMMARY AND CONCLUSIONS

Eliminating some of the gradient components thatexhibit high frequency oscillations along the thick-ness direction can enhance the performance ofplate and shell elements used in the absolute nodalcoordinate formulation. Elimination of these gradi-ent components that do not significantly affectthe solution in the case of thin plates leads tomore efficient reduced-order element that also hasbetter convergence characteristics. Two-dimensionalshape functions are used to define four-nodeelement with 36 degrees of freedom. The use ofthese shape functions allows eliminating systemati-cally the gradients defined by differentiation withrespect to the spatial coordinate along the thicknessdirection. In the reduced-order element, the bendingstiffness is accounted for using the mid-surface cur-vature. As pointed out, the use of the mid-surfacecurvature also improves the element accuracy andconvergence properties.

The good convergence characteristic of thereduced-order element is demonstrated in the case

Fig. 4 Positions of the end point A and the centre point B of the pendulum (� � �, Model I; —,

Model II)

Table 2 Simulation times as a function of the plate

thickness

Thickness (m)Ratio ofsimulation time

Simulation time (s)

Model I Model II

0.01 6 7 420.001 58.8 8 4700.0005 141.3 8 1130



Fig. 5 Position coordinates of the points A and B (– – –, Model I; ——, Model II)

Fig. 6 Deformed shape of the pendulum



of large displacement static problems. A thin cantile-ver plate was used to demonstrate the numerical effi-ciency and better convergence characteristic of thereduced-order element. It was shown that the elim-ination of the third gradient vector can have a signifi-cant effect on improving the element performance.The difference in CPU time between models basedon the high- and low-order elements was reportedfor different number of elements and thicknessvalues. The computation time was found to behighly dependent on the element thickness and dra-matically increases when the element thicknessdecreases. It was demonstrated that as the elementthickness decreases, the models based on the low-order element can be more than 100 times fasterthan the models based on elements that employfull parameterization. It is important to note thatother factors such as the integration method canhave a strong influence on the overall computationaltime. The results show that the low-order elementis computationally more efficient and owing to itsbetter convergence characteristics, fewer elementsare required when compared with the 48 degrees-of-freedom element that employs the general conti-nuum mechanics approach in the formulation ofthe elastic forces.

ACKNOWLEDGEMENT

This research was supported, in part, by the US ArmyResearch Office.

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APPENDIX 1

Notation

A rotation matrix in terms of the threeEuler angles

e vector of nodal coordinatese0 vector of nodal coordinates in the initial

configurationen vector of nodal coordinates for a node

(n ¼ A, B, C, D)E matrix of the material elastic

coefficientsi, j, k columns of the rotation matrix AI identity matrixJ matrix of the position vector gradientsM mass matrix of the elementn normal vector of the element

mid-surfaceQe external nodal forcesQk vector of the element elastic forcesr global position vector of a pointrn global position vector of a node nra vector of partial derivatives of the

position vector with respect to (a ¼ x,y, z)

raa vector of second partial derivatives ofthe position vector with respect to(a ¼ x, y)

R global position vector of the origin ofthe element coordinate system

Rn global position vector of the node due torigid body motion

Sa matrix of the element shape functions(a ¼ A, B, C)

T kinetic energy of the elementu global position vector of an arbitrary

point on the element due to rigid bodyrotations



ub position vector of a point on themid-surface in the element coordinatesystem

U strain energy of the elementVo volume of the element in the initial

curved configurationV volume of the element in the uncurved

configurationW virtual work of the element elastic

forcesx vector of the element spatial

coordinatesx, y, z spatial coordinates of the elementX, Y, Z global inertial coordinates

1xy shear strain component1xx, 1yy normal strain components in x and y

directions1 strain vector obtained from Lagrangian

strain tensor1m Lagrangian strain tensorkxx, kyy curvatures of the element mid-surface

in x and y directionskxy curvature of the element mid-surface

due to torsionk vector of curvaturesh dimensionless element coordinate in y

directionua Euler angles (a ¼ x, y, z)r material densityj position vector of the material points in

the current configurationj dimensionless element coordinate in x

direction

ja components of the global positionvector of the point in the currentconfiguration, (a ¼ 1, 2, 3)

APPENDIX 2

The shape function matrix SC used to develop thelow-order element presented in this investigation iswritten as follows

SC ¼ ½ S1I S2I S3I S4I S5I S6I

S7I S8I S9I S10I S11I S12I �

where I is a 3�3 identity matrix and shape functionsare as follows

S1 ¼ �(j� 1)(h� 1)(2h2 � hþ 2j2 � j� 1)

S2 ¼ �lj(j� 1)2(h� 1)

S3 ¼ �wh(h� 1)2(j� 1)

S4 ¼ j(2h2 � h� 3jþ 2j2)(h� 1)

S5 ¼ �lj2(j� 1)(h� 1)

S6 ¼ wjh(h� 1)2

S7 ¼ �jh(1� 3j� 3hþ 2h2 þ 2j2)

S8 ¼ 1j2h(j� 1)

S9 ¼ wjh2(h� 1)

S10 ¼ h(j� 1)(2j2 � j� 3hþ 2h2)

S11 ¼ ljh(j� 1)

S12 ¼ �wh2(j� 1)2(h� 1)

where j ¼ j=l, h ¼ y=w.



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