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Computers & Structures Vol. 19, No. 4, pp. 565-581, 1984 0045-7949/u $3.00 + .cQ Printed in the U.S.A. Pergamon Press Ltd.

MIXED FINITE ELEMENT MODELS FOR PLATE BENDING ANALYSIS:

A NEW ELEMENT AND ITS APPLICATIONS

DIMITRIS KARAMANLIDIS~

University of Rhode Island, Kingston, RI 02881, U.S.A.

HUNG LE THE$ Technical University of Berlin, Berlin (West), F.R.G.

SATYA N. ATLURI~ Georgia Institute of Technology, Atlanta, GA 30332, U.S.A.

(Received 16 August 1983; receivedfor publication 26 October 1983)

Abstract-With a few exceptions, finite element packages available in today’s commercial software environment contain in their libraries displacement-type elements only. The present paper aims to demonstrate the feasibility that properly formulated mixed-type elements compete most favorably with displacement-type elements and should, therefore, be considered as potential candidates for inclusion in general purpose finite element packages. In doing so, the development of a new triangular doubly-curved mixed-hvbrid shallow shell element and its extensive testing in carefully chosen example problems are reported on.

INTRODUCTION

Thin plates supporting transverse loading are of such common occurrence in engineering practice that the analysis of the stresses and deformations in these has attracted a considerable amount of attention over a very long period, with Refs.[21,23,24] providing much valuable information. The mathematics of the situation is so complex, however, that analytical solutions leading to formulae for stresses or deflections are only available for a few simple situations of geometry, boundary conditions, and loading. In general, progress in design work can thus only be accomplished if some approximation solution technique is used. The most versatile approach in a long list is the finite element method, which has been extensively developed, applied to many problems, and been widely discussed in the relevant literature. Several different formulations for finite element analysis of plate bending problems have been put forward and applied with success. On the other hand, however, an examination of the documentation manuals of commercial packages available on today’s software market reveals that almost exclusively a single type of element, namely the so-called assumed-displacement element, is included in their libraries. In a recently published paper by Batoz et al.[S], a comparative evaluation of several well-known thin-plate elements has been carried out. In light of the obtained numerical results, the major conclusion in Ref. [5] was that a plate element developed in the sixties on the basis of the so-called assumed stress hybrid finite element model competes most favorably with a series of elements included in commercial packages.

TAssistant Professor (formerly post-doctoral fellow, GIT-CACM).

IResearch Scientist I. $Regent’s Professor of Mechanics.

The purpose of the present paper is to demonstrate once again the practical relevance of properly formulated mixed finite elements. A newly developed mixed hybrid shallow shell element serves as the vehicle to carry out extensive numerical studies on well-selected examples. Whenever possible, the attempt is made to compare the present results with those produced by commercial-package elements. Moreover, a comparison with recently developed mixed plate elements is also made.

FORMULATION OF A NEW ELEMENT

Variational equation Customarily, mixed finite element models are for-

mulated on the basis of variational principles of the so-called Reissner type, wherein stress and displacement variables represent independent (primal) variables.

In their recent publication[l6], the authors have shown that in a mixed variational equation, choosing the stress field interpolants so as to fulfill a priori the “linear part” of equilibrium equations in the interior of each individual element leads not only to a me- chanically correct but also an easily implementable discrete scheme.

In the following, the theoretical ingredients of a newly developed shallow shell element based on such a mixed formulation will be outlined. Further documentation can be found in Ref.[21].

We consider now a free-form shell divided into (n) shallow curved triangular elements and denote by A the area, by aA the total boundary, by C, and C, the parts of the boundary over which displacement and tractions, respectively, are specified, while C, = aA - C, - Co represents the “inter-element boundary” of a single element. Furthermore, D,,, and D, are the membrane and bending parts of the

565

566 D. KARAMANLIDIS et al.

strain-stress matrices, respectively. The vectors a,,, = [N,,, N,, N,,lT and ab 5 [M,,, IV,,,,, MxJT are the shell stress resultants at a generic point in the interior of an element, while z “z(x, yj describes the element’s curved geometry with respect to its base plane (x and y are local Cartesian coordinates). In the standard vector notation, the variational equation employed within the new element’s formulation reads?

+ (T, . ii, + T3. E

+ M,, . KJ,,) dS

+ (WEF) = stationary (1)

where (WEF) stands for the work of external forces exclusive of element distributed loading and

T, = N,,+v,; T3 = M,,,p + Ncgz,&vol;

M, = M,,v, (a, /3 = 1, 2). (2)

Moreover, w and ii = [a, 6, C ] T are independent displacement fields introduced in the interior element domain A and on its boundary aA, respectively.

In eqn (l), the primal variables are subject to the subsidiary conditions listed below:

(i) C’ displacement continuity

u, +_ L&=0; w+ -3=O on C,+ _

4 - u, = 0; w - - 3=0 onC,- ‘+ -_ WY, - WY, =0 on C,

(ii) displacement boundary conditions

I&- Ii, = 0; $-$,=O _ 1 on C, w,, - I,, = 0

(iii) domain equilibrium conditions

N,,, + 15, = 0 in A M airy + (NW, . Q),. + h = 0

where F3 = j3(x, y); (i = 1,2,3) stands for the distributed loading acting on the element’s area A; (. . .)’ and (. . .)- refer, arbitrarily, to the “left-hand” and “right-hand” sides of C,, respectively; (. Y .) is the symbol for prescribed quantities; and u = Lu, U, w JT z Lu,, u2, w2 JT is the displacement vec-

tNote that in the geometrically linear case, eqn (1) degenerates to the variational basis for what one customarily calls the “assumed stress hybrid finite element method” (e.g. [51).

tor at a generic point (x, y, z) in the interior of an element.

On the other hand, taking the variation of the functional in eqn (1) with respect to the primal variables ti, a,,,, and ab gives as Euler/Lagrange equations (“natural constraints”)

(i) compatibility conditions

D,a,-y, =O D,a, - yb = 0

in A

(ii) traction reciprocity conditions

T,’ + T,- =0

T3++T,-=0 o”C,

1 M;+M,=O

(iii) C’ continuity conditions

+ - w,, - w,, = 0 on C,’

W,, - w,, - = 0 on C,-

where

u 9.x

Ym =

i I U,Y +

u,y + u,x 2

1 W,X 2

--

2

1 W,Y

2w,x.w,y

Implementation Considerable effort has been expended in the past

in developing curved shell element models. The development of such an element for thin free-form shell analysis demands that attention be paid to the following aspects of the formulation. First, one has to select a consistent shell theory from the numerous theories that have been proposed so far. Secondly, representation of elemental rigid body and constant strain modes as well as satisfaction of interelement displacement compatibility is more difficult for the curved case since the in-plane and transverse displacements are coupled due to the curvature. Thirdly, the problem of describing the geometry of the element in a proper way is encountered.

In the past, several authors, for instance Boland [8] and Wolf [28], advocated an element geometry description approach as described next. The three comer nodes of an arbitrary triangular element are chosen on the middle surface of the shell. A reference plane through these nodes is associated with

Fig. 1. Improper shell geometry approximation using curved elements.

Mixed finite element models for plate bending analysis

Table 1. Trial functions for a new mixed hybrid shallow shell element

567

2 (a and 0 . . . . . . . . . . . linear in s

cubic in s

2 o Nyy and NV . . . . . . constant plus particular solution f due to distributed loading

:M M E \ xx' yy

andM . . . . . . xy

weakly parabolic plus particular

:: solution due to distributed loading

c 2 Z..............

0 0 ( parabolic in x and y

each element. The shell element forms a shallow surface with respect to this “base-plane”. The normal projection of the element on the reference “base- plane” consists of a triangle passing through the corner nodes. Triangular elements defined this way do not cover the shell completely, as illustrated by Fig. 1. Often (see Refs. [8,28]) it has been assumed, however, that omission of the region between the elements does not affect the results in any serious way.

From the above-mentioned difficulties stems our motivation to develop a new shell element such that (i) exact representation of rigid body as well as constant strain modes is achieved, (ii) C’ interelement displacement continuity is enforced a priori, and (iii) no gaps of the aforementioned kind do occur in the element assembly. Table 1 summarizes the trial functions used within the new element’s development which comply with the subsidiary conditions of eqn (1).

NUMERICAL RESULTS

Numerical results are given in this section for a series of relevant problems representing a broad range of circumstances encountered in linear thin plate analysis. Further publications presenting results for linear and nonlinear analyses are now under preparation and will appear soon.

The problems under consideration are those for which either (i) alternative solutions, especially those obtained by commercial-type elements or, in some cases, (ii) classical solutions are available.

The first two test problems are intended to demonstrate the new element’s ability to represent exactly constant strain deformation modes. For the third example (triangular cantilever plate strip under end loading), a simple beam theory solution as well as a numerical one produced by means of a commercial element are available. The fourth class of problems deals with rectangular plates under various loading and boundary conditions. In the literature, analytical as well as numerical reference solutions are available for this type of problem. Finally, a sector plate under uniform loading is analyzed by means of the new element,

Rectangular cantilever plate strip The primary goal in this investigation was to study

the element’s performance under “single element test” situations. The plate (length L, thickness t = 0.05, width = 1.0, E = 10.0 x 106, v = 0) is divided into four elements as shown in Fig. 2. In order to study the interrelationship between the element’s aspect ratio and its performance, the length parame- ter L was increased step-wise while the other geo- metric data were held constant. The obtained numerical results for L = l-20, as summarized in Table 2, demonstrate the ability of the new element to represent exactly constant strain deformation modes (patch test!).

Even in the case of end force loading, the cor- relation between finite element and simple beam solution is very good (rel. displ. error ca. 1.3x, rel. bending moment error ca. 2.1%). It is worth noticing that the error increases as both ends of the strip are approached, and it is equal to zero in the middle of the strip. This phenomenon is well known in FEM applications and is due to the fact that in the finite element mesh used the singularities of the structure under investigation have not been taken into consideration. Further results for this example are shown in Fig. 3, in comparison with those due to Spilker and Munir[24]. A comparison between the present element and the element LH4 of Ref.[24] seems to be worthwhile for mainly three reasons. First of all, in both cases a mixed-hybrid formulation is employed. Second of all, in[24] several quadrilateral elements

Fig. 2. Cantilever plate strip-problem description.


,=

-

t&,/Pi. 1.0

75

5

25

1

2

4

5

a

10

12

14

16

18

20

Table 2. Cantibver plate st~p#mpa~soR between numerical and analytical results

End Moment

WA*

-0.0096

-0.0384

-0.1536

-0.3456

-0.6144

-0.9600

-1.3824

-1.8816

-2.4576

-3.1104

-3.8400

---$

0.0192

0.0384

0.0768

0.1152

a. 1536

0.1920

0.2304

0.2688

0.3072

0.3456

0.3840

0.0064

0.0512

0.4096

1.3824

3.2768

6.4000

1.0592

7.5615

6.2144

7.3248

1.20aa

0.0064

0.0505

0.404c

1.3646:

3.2331

6.3151

.0.9129

.7.3291

15.8678

r6.83fB

aO.5242

Ei AU

-0.0096

-0.0384

-0.1536

-0.3456

-0.6144

-0.9600

-3.. 3824

-1.8816

-2.4576

-3.1104

-3.8400

End Force

- 1.67

- 3.33

” 6.67

-10.00

-13.33

-16.67

-20.00

-23.33

-26.67

-30.00

-33.33

Mt -- - 1.56

_ 3.22

- 6.58

- 9.78

-13.04

-16.31

-19.57

-22.84

-26.09

-29.36

-32.62

,.I” A tt)) load I - endmoment i present FEM solutfon

,,,’

,.1’

i’

_ Beam theory and

present FEM solution15 elements)

1.0

9

MA0 2

- 1.00

- 2.00

- 4.00

- 6.00

- 8.00

-10.00

-12 .oo

-14 .oo

-16.00

-18.00

-20.00

_@_

-0.33

-0.67

-1.33

-2.00

-2.67

-3.33

-4.00

-4.67

-5.33

-6.00

-6.67

M: -0.44

-0.78

-1.49

-2.22

-2.96

-3.70

-4.43

-5.16

-5.90

-6.64

-7.31

,. A) ._--1 L_$“__._

,I._ ‘g.. . . ..__. -.. *c....

_ seamtheory and

prascnt FEM solw~on (2 elements)

5 75 0 25 5 75

Fig. 3. Numerical results for cantilever plate strip.

Mixed finite element models for plate bending analysis 569

Fig. 4. Anticlastic plate problem: finite element meshes.

have been proposed; and the one labeled LH4 was found to be the best among them.?

Anticlastic plate problem Like its predecessor, this problem, too, aims to

evaluate the new element’s ability of exact representation of constant strain deformation modes. Besides the analytical solution given in[26], several FEM reference solutions are available in the literature. Particular emphasis is placed here in comparing the present results with those (i) presented in[24] and (ii) obtained by finite elements included in commercial general purpose packages.

tFinally, several authors, i.e. Wunderlich[30], have claimed in the past that rectangular mixed elements perform much better than triangular ones. It will be seen that the obtained numerical results for a series of examples do by no means support this conclusion. Moreover, the superiority of the element presented here over the element LH4 of Ref. [24] is seen from Fig. 3 and will be supported by the test studies to be presented in the remainder of the paper.

A description of the problem is given in Fig. 4 together with the finite element meshes considered. For the comparison of the present results with those presented in Refs. [2,5], the following data were used: E = 104, t = 1, v = 0.3, and P = 5. Table 5 shows that, for all the meshes used, the new element pro- duces the exact solution, as does the element (labeled DKT) based on the so-called discrete Kirchhoff hypothesis[5]. This element has been implemented recently into the ADINA general purpose program[4].

Moreover, Table 3 makes clear that even upon the use of a very fine mesh (e), the solutions produced by plate elements included in STARDYNE [2] are rather inaccurate.

Further results for this case study are shown in Fig. 5 and lead again to the conclusion that the mixed- hybrid element presented here is superior to the one proposed by Spilker and Munir[24].

Triangular cantilever plate strip For this problem shown in Fig. 6, a simple beam

theory as well as a finite element solution obtained by means of the element STIF6 in ANSYS general purpose program[9] are available. Remarkably, STIF6 and the mixed-hybrid element presented here give identical results provided the same element mesh has been used. The slight difference with respect to the calculated values for the tip deflection should be attributed to the fact that unlike the present investigation, the symmetry of the problem was imposed in [9].

Rectangular plate problems A series of rectangular plates with aspect ratios

ranging from 1 to 0.5 under various loading and support conditions have been analyzed by means of the mixed-hybrid plate element presented in this paper. In addition to the standard cases of a simply supported or clamped plate under uniform or concentrated loading, the following five problems have been

Table 3. Anticlastic plate problem-comparison between numerical and analytical results

ey _ 03,*o ( . . . . . . . . . . ( __.......... . . . . . . . . . . . . ( . . . . . . . . . . . . ( . . . . . . . . . . . . . . . . . . . . . . . . _.____ __ .___ 1 __._._._____

Mu o 1 , f

00’ -- . ..-.... -- . . . . - ..-... _ . . . . . . . . .._. .- . . . . . . . . . . ________ ____ __.____ _____ ____. _______ i o.

2 s . . . . . .._.... ._.......... . . . . . . . . . . . . . . . . . . . . . . . . . .._ . . . . . . . . . . . . . . . . . . . . . . ..___...._ 5

“g 00

I I s t _

, ------------ , . . . . . . . . . , . . . . . _ _ _ _ _ , . . . . . . . . . . . . , . . . . . . . . . . . , . . . . . . . . . . . . ,._____....__ El ,” *5m ,

D. KARAMANLILXS et al.

Plate theory and I present FEM solution 12 elements) 1

Uniform Mq 50

WLKER /MUNlR I (16 element 51

wthln 2 % wIthIn 3 % ’

Fig. 5. Comparison of numerical results for the anticlastic plate problem.

Beam theory -.0126666 200.0

ANSYS - ,012 666 8 200.0

present solution -.OL26678 200.0

Fig. 6. Triangular cantilever plate strip.

CASE I


l------T T------ CASE IL? CASE P

LL---________J -------_-_

- : clamped

-__- : simply supported

El ’ point support

Fig. 7. Rectangular plate under uniform loading-cases I-V.

investigated (see Fig. 7): (i) In all cases, only a relatively coarse mesh (i) two opposite sides clamped and the remaining

simply supported, (ii) two opposite sides simply supported and the

remaining free, (iii) point supports at the corners, (iv) two adjacent sides simply supported and point

support at the fourth corner, and (v) one side simply supported and point supports

at the opposite comers. We consider first the cases of a simply supported

or clamped rectangular plate under concentrated or uniform loading. Tables 5-9 and Figs. 8-10 show the numerical results for a square plate predicted by the new element as well as by other well-established elements. Similarly, Tables 10 and 11 summarize results for the case of a plate with an aspect ratio of 1:2. On the basis of these results, the following conclusions can be made:

(4 x 4) is needed in order to achieve, by the new element, a solution accuracy which is sufficient for practical purposes. This is not only true for the displacement but also (and most importantly) for the bending moment field.

(ii) It can be argued that other elements, too, produce results of comparable or even better accuracy than the element presented in this paper. It seems, however, that the following facts favor the latter one: (a) Application of element B-21 on practical situations is prohibited by the large number of DOF per element as well as by the superfluously imposed C2 compatibility; (b) Element Z is irrelevant due to its mathematical deficiencies; (c) Hybrid displacement elements (like HK and KA) can under certain (realistic) circumstances become numerically unstable; (d) The number of DOF per element in the cases of KDKT and EQT is relatively large; (e) The

Table 4. List of elements considered in square plate analysis problems

Notation

?I

E’?T

HCT

KDKT

BDKT

KA

HK

7.

B-21

FEM Approach

displacement

force

displacement

displacement (pseudo mixed)

displacement

hybrid displacement

hybrid displacement

displacement

displacement

Number of DOF Author

9 Martin (STARDYNE)

16 de &&eke/Sander

12 Bsieh/Clough/Tocher (STARDYNE)

12 Kikuchi

9 BatOz/Bathe/nO (ADINA)

9 Kikuchi/Ando

9 Harvey/Kelsey

9 Bazeley et al. (ASAS)

21 Bell

512 D KAIMfANLIDIS et al.

Table 5. Simply supported square plate under concentrated load

PRESENT WORK KDKT BDKT EQT HK B-21

Mesh A -I---+ 9.83796 1x1

(-16.2%)

INALYTICAL 11.6008

I

(-0.9%) (+0.295%) (+3.44%) (+0.6%) (-2.5%) (-0.6%)

MULTIPLIER ( P.(2d2 / D ).103 I

Table 6. Clamped square plate under concentrated load

PRESENT WORK KDKT BDKT EQT HCT z

Mesh A Mesh B Mesh A Mesh B Mesh A Mesh B Mesh B Mesh B Mesh B

2.604167 1.736111 2.285192 2.285192 5.699 6.219 8.2565 1.0 5.21

1X1 (-53.5%) (-69.0%) (-59.23%) (-59.23%) (+ 1.23%) (+11.05%) (+47.31%) (-82.16%) (-7.05%)

5.022159 4.685122 5.080605 5.041774 5.855 6.360 6.1939 4.2400 5.89

g 2x2

k! (-10.4%) (-16.4%) (-9.36%) (-10.05%) (+4.56%) (+13.57%) (+10.51%) (-24.35%) (+5.09%)

": w 5.440736 5.348003 5.517820 5.498133 5.707 5.911 5.7551 5.192 5.72

4x4 (-2.93%) (-4.59%) (-1.56%) (-1.90%) (+1.92%) (+5.54%) (+2.69%) (-7.37%) (+z.os%)

ANALYTICAL SOLUTION 5.605

MULTIPLIER I ( P.(2d2 / D ).103


Table 7. Simply supported square plate under uniform loading

/-- - _ _ _ _ _ _ _ _

/

// /

/

~~~

/I /

/I / /q p

I

L__.____:___~ 1 I-+---- 2a ----I

PRESENT WORK KDKT

Mesh A Mesh B Mesh A Mesh B

3.279321 1.591435 4.166667 3.703768 1x1

(-19.28%) (-60.82%) (+2.57%) (-8.83%)

3.862935 3.426549 4.073719 4.019489

~~

BKDT

Mesh A

4.161

(+2.49%)

4.056

(-0.1%) --

4.065

(+0.12%)

I ANALYTICAL SOLUTION I 4.062353

HK KA

Mesh A Mesh A Mesh B

4.407 3.627

(+8.48%) (-10.72%)

4.10 4.092 4.081

(0.93 %) (+0.73%1 (+0.46%)

4.06 4.069 4.074

(-0.06%) (+U.16%) (+0.29%)

MULTIPLIER ( q.(2aJ4 / D )*103

disadvantages of BDKT when compared with the element presented here are a less accurate stress field prediction combined with its sensitivity with respect to mesh orientation.

(iii) The superiority of the new element when compared with the recently developed rectangular mixed-hybrid element LH4 of Ref. [24] as well as with the element M included in the STARDYNE general purpose finite element program[2] is evidenced once again by the results presented in Figs. 9, 10, 12 and 13.

In Table 12, numerical and analytical results for a square plate under uniform loading and subject to boundary conditions corresponding to the aforementioned cases I-III are summarized. Again, excel- lent agreement between the results predicted by the new element and the analytical or numerical results of Refs.[l4,21] is demonstrated. Figures 1416 aim to show how a square plate subject to the aforementioned boundary conditions deforms upon the action of uniform loading.

In the final part of this study, several rectangular plates having an aspect ratio ranging from 1.0 to 0.5 and subject to boundary conditions corresponding to the aforementioned cases III-V have been analyzed by means of the new mixed-hybrid element. In all cases, uniform loading has been considered and an (8 x 8)A finite element mesh (Fig. 8) has been used Again, very good agreement between the present FEM results and the analytical ones is demonstrated.

Sector plate under uniform loading The sector plate under uniform loading with two

adjacent sides fixed and the other ones left free (see Fig. 17) has been analyzed previously by Knothe [2 11. In that paper, a rectangular element was developed on the basis of the classical force method. The predicted numerical solution by the element presented in this paper is in very good agreement with the one reported in[25]. This is remarkable due to the fact that in[25] a specially tailored procedure (incor- poration of the boundary and interelement traction continuity conditions in the elemental trial functions, symmetry with respect to the diagonal, etc.) was adopted, while in this paper the problem was treated without taking advantage of its special features. Moreover, a uniform (6 x 6)A mesh was used, which, obviously, is by no means the most appropriate mesh to treat this problem (singularities!).

SUMMARY AND CONCLUSIONS

A triangular shallow curved element for the elastic analysis of thin free-form plates and shells has been presented. The new element’s formulation is based on a mixed variational equation wherein stress and displacement variables represent the independent (primal) variables. Efficiency, reliability, and accuracy of the new element have been demonstrated by a series of well-selected examples covering a broad range of thin plate analysis. Despite what is customarily believed, the obtained numerical results lead to

574 Table 8. Clamped square plate under uniform loading

PRESENT WORK KDKT BDKT KA

Mesh A Mesh B Mesh A Mesh B Mesh A Mesh A Mesh B

0.868056 0.289352 0.723644 0.418951 1.889 1.042 0.372 1x1

(-31.4%) (-77.1%) B

(-42.81%) (-66.89%) (+49.97%) (-17.65%) (-70.6%)

P 1.317030 0.987463 1.212608 1.162063 1.547 1.288 1.113

ANALYTICAL

SOLUTION

MULTIPLIER

1.26532

( q.(2d4 / D ).103

I

ANALYTICAL

SOLUTION

MULTIPLIER

Table 9. Square plate under uniform loading-evaluation of bending moments

SIMPLY SUPPORTED CLAMPED

M:, M& di MfY &'x Mty M,cx Miy

0.0604 0.0539 0.0551 0.0198 0.0344 0.0344 0.2315E-3 1.46~-2

+26.15%) (+12.58%) (-15.23%) (-14.32%) (+48.81%) (-33.0%)

0.0409 0.0447 0.0809 0.0314 0.0316 0.0473

0.20463-3 0.9123-3 l-14.72%) (-6.79%) (+24.46%) (+35.93%) (+36.80%) (-7.75%)

0.0487 0.0482 0.0641 0.0250 0.0247 0.0495

0.04043-3 0.258E-3 :+1.62%) (+0.68%) (-1.46%) (+8.06%) (+6.52%) (-3.46%)

0.0479 0.0479 0. 0.065 0.0231 0.0231 0.0513 0.

¶.(2d2


mesh A mesh B

Fig. 8. Finite element meshes used for the analysis of various rectangular plate problems.

*MO-

-rc.-. -.-._ -““---e...-” . . . . . .._..“._.-.*

6 7 8 tlel

. . . . . . . . . * simply supper t ed

present solution

a,r : Spllker/Munir

Fig. 9. Square plate under concentrated loading: error in deflection at center.

Fig. 10. Square plate under uniform loading: error in deflection at center.


Table 10. Simply supported rectangular plate under concentrated load

/r---- __----_--

YP I

SOLUTION 16.5239

MULTIPLIER ( P.(2aj2 / D ). lo3

Table 1 I. Clamped rectangular plate under concentrated load

ANALYTICAL 7.215

SOLUTION

MULTIPLIER ( P-G-d2 / D ).103


Fig. 11. Bending moment distribution for a square plate under uniform loading.

‘17

i\

\t

4. ,. _ !” ‘.. ! ‘\

f

t40.0

+30.0

t2O.c

tKx

C

-10.

-20.

-30.

-40.

-50

)- I

0

0 ii

simply supported

8,B mesh A pesent rolutiwl

O,+ mesh 8

A,V mesh 8 Martin

0,. mesh 8 de Veubeke/Sandsr

Fig. 12. Rectangular plate under concentrated loading: error in deflection at center.

Y

cm

f

+40x

+3clo

+2m

+10.0

0

-10.0

-2o.c

-3o.c

-4o.c

-5o.c ‘I %

I

I

\ ‘\ \. \ \ ,

\.

f =

w

fe;;x

~cyx

ac~

k... .

. 1.

. .

a-.. .

. . . .._

_.._

. *,

‘\.

1.

~

~‘V

_.._

_ L.

“‘...,

, =

-v

.‘X.,_

fi-..

-.-.

. . .

. . .

. . .

. ...

” .

. .

. .

. ..

__

.__

....

._._

....

....

. _

__

,,,,

_ o

e

-.-

clam

ped

q

U

mes

h

A

pre

sen

t so

luti

on

0

+ m

esh

B

D

V

mes

h

B

Mar

tin

1.

d :

i

).

Fig.

13

. R

ecta

ngul

ar

plat

e un

der

unif

orm

lo

adin

g:

erro

r in

def

lect

ion

at c

ente

r.

Tab

le

12.

Squa

re

plat

e un

der

unif

orm

lo

adin

g-nu

mer

ical

re

sults

fo

r ca

ses

I-II

I

L 20

A

case

I

case

II

Cas

e III

anal

ytic

al

solu

tio

n

Kan

t

Knothe

b

multiplier

( a

. ..)

(4 x 4)A (

C...) b

(6 x 6)

i Finite Element Mesh (Quadrant)

0.33

613-

2

0.85

05E

-2

I

-_

.-

.-

.-

.,--

_ .

. .

. ..-

.-


Table 13. Rectangular plate under uniform loading (cases III-V)

4 q a /rL..for cases (iii) & (iv)

;= I 5 q a /D...for case (v)

b/a 1.0 0.9 0.8 1 0.7 7 0.6 1 0.5 1 / I I I I , I

cl I* 0.0263 0.0218 0.0180 0.0158 0.0148 0.0140

iA 0.0256 0.0212 0.0180 0.0145 0.0137

( ).: present FEM solution

undeformed state

deformed state

Fig. 14. Square plate under uniform loading: case I.

undeformed state

Fig. 16. Square plate under uniform loading: case III.

undeformed state

v\

undeformed state

eformed state

eformed stote

Fig. 15. Square plate under uniform loading: case II. Fig. 17. Deformed and undeformed geometry of an angular

plate.

580

Fig. 18. Sector plate under uniform loading: bending moment distribution.

the illusion that the new mixed-hybrid element competes most favorably when compared with bath commercial package elements and rectangular mixed elements. It is believed, therefore, that properly formulated mixed-hybrid elements deserve a better treat- ment by general purpose program developers and should be considered as potential candidates for inclusion in the same.

Acknowledgements-This work was carried out with financial assistances from the Research Council of the Technical University of Berlin (FNK) and the German Science Foundation (DFG) to the first author under’Grants FPS 9/2 and Ka 487/3. The authors also acknowledge partial support provided by the Georgia fnstitute of Tech- noiogy. Last but not least thanks are extended to Ms. J. Webb for her assistance in preparing the manuscript.

RRPBRgNCES f * Anonymous, PAFEC IS-Theory, Results. Nottingham

University (1975). 2. contour M~/STA~~E -finite eIement

d~~s~t~~n problems. Control Data Corporation, Minnesota (1973).

3. S. N. Atluri and T. H. H. Pian, Theoretical formulation of finite-element methods in linear-elastic analysis of genera1 shells. J. Strucr. Mech. 1, 1-41 (1972).

4. K. f. Bathe, ADINA-a finite element program for automatic dynamic incrementi nonlinear analysis, Acoustics and Vib~tion Lab. RePort 82448-l. Dept. of Mechanical Engineering, M&T., Sept. 1975 (revised May 1977).

5, J.-L. Batoz, K.-J. Bathe and L.-W. Ho. A study of three-node triangular plate bending elements. Znt. 1. Numor. Meth. Engng 25, 1771-1812 (1980)

6. G. P. Bazeley, Y. K. Cheung, B. M. Irons and 0. C, zienkiewicz, Trianguhtr etements in plate bending- confo~ng and non~nfo~ng solutions. Proc. Co@ on Matrix Methods in Strucitiral Mechanics, pp. 399-440. WPAFB, Ohio, (1968).

7. K. Bell, A refined triangular plate bending finite element. In?. J. Numer. Meth. Engng 1, 101-122 (1969).

8. P. L. Boland, Large deflection analysis of thin elastic structures by the assumed stress hybrid finite element method. Thesis presented to the Massachusetts Institute of T~hno~o~, at Cambridge, Massachusetts, in 1975, in partial fulfillment of the requirements for the degree of -Doctor of Philasophy. _

9. C. J. DeSalvo. ANSYS enaineerine analysis svstem verification manual. Swanso~Analy& Systems (i976).

10. B. F. DeVeubeke, Displacement and equilibrium models in the finite element method. Stress An&@ (Edited by 0. C. Eienkiewfcz and G. S. Holster). Wiley, _. _ .

23. R. J. Roark and W. C. Young, Formulas for Stress and Strain, 5th l?.dn. McGraw-Hill, New York (1975).

24. R. L. Spilker and N. I. Munir, The hybrid stress model for thin plates. ZRt. J. Nmer. Me&. Engng 15, 12391260 (1930).

25. R. Szilard, Theory and Analysis of Plates fClassictd attd Numerical Methuds). Prentice-Hall, Englewood Cliffs, New Jersey (1974).

26. S. P. Timoshenko and S. Woiaowsky-Krieger, Theory of Plates and Shells, 2nd Edn. McGraw-Hill, New York (1959).

27. U. Wakier, FLASH-A simple tool for complicated Chichester (1966). problems. Adurmces Engng Ssftwure 1, 137-140 (1979).

11. R. H. Gatfagher, Problems and progress in thin shelf finite element analysis. Bite Bements for TIdn Shells and Curved Members (Edited by D. G. Ashwell and R. H. Gallagher). Wiley, New York (1976).

12, J. W. Harvey and S. Kelsey, Triangular plate bending elements with enforced compatib’llity. ‘AZAA J. 9; 102~1026 (1971).

14.

15

16.

k3. G. Ho&&toe, Finite efement instabihtv am&is of free-form shells. Report No. 77-2, Unive&ty of Tron- heim (1977). T. Kant, Numerical analysis of thick plates. Comput. Meth. Appl. Me&. Engng 31, l-18 (1982). D. Karamanhdis, Beitragzur Iinearen und nichtlinearen Elastokinetik der Systeme und Kontinua (Theorie- ~r~hnung~~~i~An~~~i$piele). Habihta- tion thesis, Tech&a1 University of Berlin (1983). D. Karamanlidis, A new mixed hybrid finite element model for static and dynamic analysis of thin plates in bending. Proc. ASCE EMD Specialty Conf. West La- fayette, 23-25 May (1983). D. Karamanlidis and S. N. Atluri, Mixed finite element models for plate bending analysis: theory. Paper sub- mitted for publication (May 1983). F. Kiuchi and Y. Ando, A new variationai functiord for the finite element method and its application to plate and shell problems. Nucl. Engng Lh.@n 21, 95-113 (19721. F. ckuchi and Y. Ando, Some finite element solutions for plate bending problems by simplified hybrid displacement method. Natal. Engng Design 23, 155-178 (1972). F. Kikuchi On a mixed method related to the discrete Kirchhoff assumption. Hybrid and Mixed Finite Ele- ment Models (Edited by S. N. Atluri et al.). Wiley, New York (1983).

17.

18.

19.

20.

21. K. Knothe, Plattenberechnung nach dem Kraftgriissen- verfahren. Der Stah~b~u 36. 202-214 and 234-254 f1%7j.

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APPENDIX On Kikuchi’s “mixed” model

In a recent pubhcation[20], Kikuchi proposed a triangular thin plate element with 12 DOF which is based on the so-called discrete Kirchhoff hypothesis. As it should become apparent from the discussion to follow, despite Kikuchi’s choice to label his element “mixed”, it seems, however, that the theoretical concept of this element is very closely related to the standard assumed displacement finite element approach. As a matter of fact, the variational equation employed in[20]

1 dA

-lb.w.dA+~“~~~.(w,,+e~) + A,. 6$ + e,)] dA

I

= stationary (3)

is nothing other than a modified principle of stationary potential energy. This formulation can be obtained from the standard one, see eqn (30) of Ref. [17], by replacing in the functional the lateral displacement derivatives w,, and w,~ by the rotations 0, and t?,, respectively, and relaxing the kinematic (so-called KirchhoB) constraints

w,, + e, = 0; w,~ + e, = 0

by means of the Lagrangian multipliers AX and A,.

Taking the variation of the functional in eqn (3) with respect to its primal variables, we obtain the natural constraints

se,: - D 4, + e,,, + q (e,, + + 2x = 0 (4a)

60,: - D e,, + exFy + 7 ce,., + + 5 = 0 (4b)

2x,X + 5, + p = 0 (4c)

w,, + e, = 0 (4d)

w,~ + e, = 0. (4e)

By means of eqns (4a)-(4c), the Lagrangian multipliers AX and Ay can be identified as the plate shear forces. Using eqns (4a) and (4b) in order to eliminate A, and A? from eqn (3) leads to the variational equation employed by Batoz et al. [S]. Instead of doing so, in [20] A, and 1, have been treated as independent variables. Thus, at that point an element developed on the basis of eqn (3) is indeed a mixed one. When compared, however, with “standard” mixed elements based on Reissner-type variational principles, the following oerolexitv of Kikuchi’s element (called KDKT in the follow- . I

&g) becomes apparent. Within‘this element concept, shear forces are treated as independent variables, while the bending moments M,, Myy and Mv are dependent variables. Certainly, this feature of the KDKT element stands in contradiction with the Kirchhoff thin plate theory.

In[20] an error estimation of the proposed finite element model was presented and the major conclusion made that the accuracy of the KDKT element is comparable to that of the conforming HCT element. It should be pointed out, however, that only the results for displacement quantities but not for stress resultants have been proposed. (Note that within KDKT the bending moments are calculated in exactly the same way as within a displacement-type element.) Therefore, in our opinion there is no evidence that the “mixed” KDKT element with 12 DOF has to offer any advantage when compared with its displacement-type coun- terpart BDKT element (20) with nine DOF.