Computers & Structures Vol. 19, No. 4, pp. 565-581, 1984
Printed in the U.S.A.
0045-7949/u $3.00 + .cQ
Pergamon Press Ltd.
MIXED FINITE ELEMENT MODELS FOR
PLATE BENDING ANALYSIS:
A NEW ELEMENT AND ITS APPLICATIONS
DIMITRISKARAMANLIDIS~
University of Rhode Island, Kingston, RI 02881, U.S.A.
HUNG LE THE$
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 publication26 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
GIT-CACM).
IResearch Scientist I.
$Regent’s Professor of Mechanics.
fellow,
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 formulated 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 mechanically 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
D.
566
et al.
KARAMANLIDIS
strain-stress
matrices, respectively. The vectors
a,,, = [N,,, N,, N,,lT and ab5 [M,,, IV,,,,,MxJTare 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?
tor at a generic point (x, y, z) in the interior
element.
On the other hand, taking the variation
functional in eqn (1) with respect to the
variables ti, a,,,, and ab gives as Euler/Lagrange
tions (“natural constraints”)
of an
of the
primal
equa-
(i) compatibility conditions
D,a,-y,
=O
D,a,
0
-
yb =
in
A
(ii) traction reciprocity conditions
T,’ + T,- =0
T3++T,-=0
M;+M,=O
o”C,
1
(iii) C’ continuity conditions
+
(T, . ii, + T3. E
+ M,, . KJ,,)dS
+
w,, - w,, = 0
on
C,’
- = 0
W,, - w,,
on
C,-
where
+ (WEF) = stationary
u 9.x
(1)
where (WEF) stands for the work of external forces
exclusive of element distributed loading and
T, = N,,+v,;
M,
I
1
+
U,Y
Ym =
i
u,y
+
1
u,x
2
W,X
2
T3 =
= M,,v,
M,,,p
+ Ncgz,&vol;
(a, /3 = 1, 2).
--
w+ -3=O
on C,+
onC,w - - 3=0
‘+
-_
=0
on C,
WY,- WY,
(ii) displacement boundary conditions
$-$,=O
I&- Ii, = 0;
2w,x.w,y
(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;
_
4 - u, = 0;
W,Y
2
1
on C,
_
w,, - I,, = 0
(iii) domain equilibrium conditions
N,,, + 15,= 0
in A
M airy + (NW,. Q),. + h = 0
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
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
and
quantities;
prescribed
symbol
for
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).
Fig.
1. Improper
shell geometry approximation
curved elements.
using
Mixed finite element models for plate bending analysis
567
Table 1. Trial functions for a new mixed hybrid shallow shell element
2 (a and 0
. . . . . . . . . . . linear in s
cubic in s
2o
Nyy
and NV
. . . . . . constantplus
Myy
andM
. . . . . . weakly parabolic plus particular
solution due to distributed loading
particular solution
due to distributed loading
f
:M
\
E
::
c
2
0
0
xx'
xy
(
Z..............
each element. The shell element forms a shallow
surface with respect to this “base-plane”. The normal
projection of the element on the reference “baseplane” 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,
parabolic in x and y
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 parameter L was increased step-wise while the other geometric 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 correlation 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.
568
D.
Table 2. Cantibver plate st~p#mpa~soR
End
between numerical and analytical results
Moment
End
WA* ---$
1
,=
et al.
KARAMANLIDIS
-0.0096
Force
AU
0.0192
0.0064
MA0 _@_
-- Mt
Ei
0.0064
-0.0096
-
1.67
-
M:
2
1.56
-
1.00
-0.33
-0.44
2
-0.0384
0.0384
0.0512
0.0505
-0.0384
-
3.33
_
3.22
-
2.00
-0.67
-0.78
4
-0.1536
0.0768
0.4096
0.404c
-0.1536
”
6.67
-
6.58
-
4.00
-1.33
-1.49
9.78
5
-0.3456
0.1152
1.3824
1.3646:
-0.3456
-10.00
-
-
6.00
-2.00
-2.22
a
-0.6144
a.
3.2768
3.2331
-0.6144
-13.33
-13.04
-
8.00
-2.67
-2.96
10
-0.9600
0.1920
6.4000
6.3151
-0.9600
-16.67
-16.31
-10.00
-3.33
-3.70
12
-1.3824
0.2304
1.0592
.0.9129
-3..
-20.00
-19.57
-12
.oo
-4.00
-4.43
14
-1.8816
0.2688
7.5615
.7.3291
-1.8816
-23.33
-22.84
-14
.oo
-4.67
-5.16
16
-2.4576
0.3072
6.2144
15.8678
-2.4576
-26.67
-26.09
-16.00
-5.33
-5.90
18
-3.1104
0.3456
7.3248
r6.83fB
-3.1104
-30.00
-29.36
-18.00
-6.00
-6.64
20
-3.8400
0.3840
1.20aa
aO.5242
-3.8400
-33.33
-32.62
-20.00
-6.67
-7.31
1536
3824
-
I present
i
A tt))load
- endmoment
,.I”
,,,’
FEM solutfon
,.1’
i’
t&,/Pi.
1.0
_
Beam
present
theory
1.0
and
FEM solution15
._--1
,. A)
L_$“__._
‘g..
. . ..__.
-..
*c....
,I._
elements)
75
_
seamtheory
prascnt
9
and
FEM solw~on (2 elements)
5
25
5
75
0
25
Fig. 3. Numerical results for cantilever plate strip.
5
75
569
Mixed finite element models for plate bending analysis
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 produces the exact solution, as does the element (labeled
DKT) based on the so-called discrete Kirchhoff
hypothesis[5]. This element has been implemented
into
the ADINA
general
purpose
recently
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 mixedhybrid element presented here is superior to the one
proposed by Spilker and Munir[24].
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.
Table 3. Anticlastic plate problem-comparison
ey
Mu
s
5
I
03,*o
1,
f
2
I
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
between numerical and analytical results
( .... ...... ( __...................... ( ............ ( ............ ............
_
o
Triangular cantilever plate strip
00’
-- . ..-....
--
. . . . - ..-...
_
. . . . . . . . .._.
.- . . . . . . . . . .
________ ____
__.____ _____
_.____
__
.___
1__._._._____
____. _______
i
o.
“g
.. .. ..._.... ._.......... .. .. .. .. .. .. .. .. .. .. .. .. ..._.. .. .. .. .. .. .. .. .... .. ..___...._
st
00
_
,”El *5m ,
, ------------, .. .. .. . .. , . .. .. _____, .. .. .. .. .. .. , .. .. . .. .. .. , .. .. .. .. .. .. ,._____....__
D.
Plate
present
theory and
FEM solution
WLKER
/MUNlR
(16 element 51
I Uniform
1
12 elements)
wthln
et al.
KARAMANLILXS
2 %
I
wIthIn
Mq
50
3 %
’
Fig. 5. Comparison of numerical results for the anticlastic plate problem.
Beam
theory
ANSYS
present
solution
-.0126666
200.0
- ,012 666 8
200.0
-.OL26678
200.0
Fig. 6. Triangular cantilever plate strip.
Mixed finite element models for plate bending analysis
CASE
I
l------T T-----CASE
CASE IL?
LL---________J
P
-------_-_
-__El
:
clamped
:
simply
supported
’ point support
Fig. 7. Rectangular plate under uniform loading-cases
investigated (see Fig. 7):
(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:
I-V.
(i) In all cases, only a relatively coarse mesh
(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
FEM Approach
Number
of
?I
displacement
E’?T
force
16
HCT
displacement
12
KDKT
BDKT
displacement
(pseudo
mixed)
displacement
9
DOF
Author
Martin
de
Bsieh/Clough/Tocher
(STARDYNE)
Kikuchi
12
9
(STARDYNE)
&&eke/Sander
BatOz/Bathe/nO
(ADINA)
KA
hybrid
displacement
9
Kikuchi/Ando
HK
hybrid
displacement
9
Harvey/Kelsey
7.
displacement
9
B-21
displacement
21
Bazeley
et
Bell
al.
(ASAS)
512
D
KAIMfANLIDIS
et al.
Table 5. Simply supported square plate under concentrated load
KDKT
PRESENT WORK
BDKT
EQT
HK
B-21
(-2.5%)
(-0.6%)
-I---+
Mesh A
9.83796
1x1
(-16.2%)
(+0.295%) (+3.44%)
(-0.9%)
(+0.6%)
INALYTICAL
11.6008
MULTIPLIER
( P.(2d2
I
/ D ).103
I
Table 6. Clamped square plate under concentrated load
PRESENT WORK
BDKT
KDKT
EQT
Mesh A
Mesh B
2.604167
1.736111
(-53.5%)
(-69.0%)
5.022159
4.685122
5.080605
(-10.4%)
(-16.4%)
(-9.36%) (-10.05%) (+4.56%)
5.440736
5.348003
5.517820
5.498133
5.707
(-2.93%)
(-4.59%)
(-1.56%)
(-1.90%)
(+1.92%) (+5.54%)
Mesh A
Mesh B
Mesh A
Mesh
2.285192
2.285192
5.699
6.219
B
Mesh B
8.2565
HCT
Mesh B
1.0
z
Mesh B
5.21
1X1
g
k!
":
w
(-59.23%)
(-59.23%)
(+
5.041774
1.23%) (+11.05%) (+47.31%) (-82.16%) (-7.05%)
5.855
2x2
4x4
ANALYTICAL
SOLUTION
MULTIPLIER
6.360
5.911
5.605
I
(
P.(2d2
6.1939
4.2400
5.89
(+13.57%) (+10.51%) (-24.35%) (+5.09%)
/ D ).103
5.7551
(+2.69%)
5.192
(-7.37%)
5.72
(+z.os%)
573
Mixed finite element models for plate bending analysis
Table 7. Simply supported square plate under uniform loading
- _ _ _ _ _ _ __
/
/
/
/-//
/I
/q
/I
~~~/
/
p
I
1
L__.____:___~
I-+---PRESENT WORK
2a ----I
KDKT
BKDT
HK
Mesh A
Mesh A
Mesh B
Mesh A
Mesh B
Mesh A
3.279321
1.591435
4.166667
3.703768
(-19.28%) (-60.82%)
(+2.57%)
4.073719
KA
Mesh A
Mesh B
4.161
4.407
3.627
(-8.83%)
(+2.49%)
(+8.48%) (-10.72%)
4.019489
4.056
1x1
3.862935
3.426549
(-0.1%)
--
~~
ANALYTICAL
I
SOLUTION
MULTIPLIER
(0.93 %)
4.092
4.081
(+0.73%1
(+0.46%)
4.065
4.06
4.069
4.074
(+0.12%)
(-0.06%)
(+U.16%)
(+0.29%)
4.062353
I
( q.(2aJ4 / D )*103
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, excellent 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.
disadvantages
4.10
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 (incorporation 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
KDKT
PRESENT WORK
BDKT
KA
Mesh A
Mesh B
0.868056
0.289352
0.723644
(-31.4%)
(-77.1%)
(-42.81%) (-66.89%) (+49.97%) (-17.65%) (-70.6%)
1.317030
0.987463
1.212608
Mesh A
Mesh B
0.418951
Mesh A
Mesh A
1.889
Mesh B
1.042
0.372
1x1
B
P
1.162063
1.547
1.288
1.113
ANALYTICAL
1.26532
SOLUTION
MULTIPLIER
( q.(2d4
/ D ).103
Table 9. Square plate under uniform loading-evaluation
of bending moments
SIMPLY SUPPORTED
M:,
0.0604
M&
di
0.0539
CLAMPED
MfY
0.0551
&'x
0.0198
Mty
M,cx
0.0344
0.0344
0.2315E-3
+26.15%)
0.0409
(+12.58%)
1.46~-2
(-15.23%) (-14.32%) (+48.81%) (-33.0%)
0.0447
0.0809
0.0314
0.0316
0.0473
0.20463-3
l-14.72%) (-6.79%)
0.0487
0.9123-3
(+24.46%) (+35.93%) (+36.80%) (-7.75%)
0.0482
0.0641
0.0250
0.0247
0.0495
(-1.46%)
(+8.06%)
(+6.52%)
(-3.46%)
0.065
0.0231
0.0231
0.0513
0.04043-3
I
:+1.62%)
(+0.68%)
0.0479
0.0479
Miy
0.258E-3
ANALYTICAL
SOLUTION
MULTIPLIER
0.
¶.(2d2
0.
575
Mixed finite element models for plate bending analysis
mesh A
mesh B
Fig. 8. Finite element meshes used for the analysis of various rectangular plate problems.
*MO-
-rc.-.
-““---e...-”
-.-._
. .. . . .._..“._.-.*
7
6
.. .. .....*
simply
tlel
supper t ed
present
a,r
8
solution
: 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.
D.
576
et al.
KARAMANLIDIS
Table 10. Simply supported rectangular plate under concentrated load
__----_-/r---IYP
SOLUTION
MULTIPLIER
16.5239
( P.(2aj2
/ D ). lo3
Table 1I. Clamped rectangular plate under concentrated load
ANALYTICAL
SOLUTION
MULTIPLIER
7.215
( P-G-d2
/ D ).103
Mixed finite element models for plate bending analysis
Fig. 11. Bending moment distribution for a square plate under uniform loading.
‘17
f
i\
t40.0
\t
+30.0
,.
!”
! ‘\
4.
_
‘..
t2O.c
tKx
C)I
-10.0
simply supported
-20.
8,B
mesh A
O,+
mesh 8
pesent rolutiwl
-30.
A,V 8
0,.
-40.
-50 0
mesh
Martin
mesh 8
de Veubeke/Sandsr
ii
Fig. 12. Rectangular plate under concentrated loading: error in deflection at center.
I
,
‘I %I
d
: i
\
U mesh A
D V mesh B
‘\.
1.
. . . . . . ..__.__....._._.........
.
L.
Martin
present solution
.
_ __,,,,_
o
=-v
f = wfe;;x~cyxac~
clamped
. . . . . . . . . ...”
0 + mesh B
q
fi-..-.-..
1.
k.... .
-.-
\.
.‘X.,_
\.
\
‘\
~~‘V_..__
“‘...,,
a-........__.._.
*,
\
e
-_
.-
.-
.-
.,--
Fig. 13. Rectangular plate under uniform loading: error in deflection at center.
-5o.c).
1.
-4o.c
-3o.c
-2o.c
-10.0
0
+10.0
+2m
+3clo
+40x
f
_
b
II
I
Knothe
Kant
analytical
solution
(6 x 6)
i
(4 x 4)A (
multiplier
III
a
C...)
( . ..)
Case
case
case
L
b
I
.
. .
.
Finite Element Mesh (Quadrant)
20 A
Table 12. Square plate under uniform loading-numerical
..-
.-
0.8505E-2
0.33613-2
results for cases I-III
Y
cm
579
Mixed finite element models for plate bending analysis
Table 13. Rectangular plate under uniform loading (cases III-V)
/
I
0.8
0.9
I
I
1
0.7
7
I
I*
0.0263
0.0218
0.0180
iA
0.0256
0.0212
0.0180
0.6
,
0.0158
1
0.5
1
I
0.0148
0.0140
0.0145
0.0137
q
I
a
;=
1.0
b/a
cl
4
/rL..for cases (iii) & (iv)
5
q a /D...for case (v)
( ).:
present FEM solution
undeformed
undeformed
deformed
state
state
Fig. 14. Square plate under uniform loading: case I.
undeformed
state
Fig. 16. Square plate under uniform loading: case III.
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 treatment 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 Technoiogy. 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).
eIement
2. contour
M~/STA~~E
-finite
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. RePort82448-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 bendingconfo~ng
and non~nfo~ng
solutions. Proc. Co@
on MatrixMethodsin StrucitiralMechanics, 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,
Chichester (1966).
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).
k3. G. Ho&&toe, Finite efement instabihtv am&is of
free-form shells. Report No. 77-2, Unive&ty of Tronheim (1977).
14. T. Kant, Numerical analysis of thick plates. Comput.
Meth. Appl. Me&. Engng 31, l-18 (1982).
15 D. Karamanhdis, Beitragzur Iinearen und nichtlinearen
Elastokinetik der Systeme und Kontinua (Theorie~r~hnung~~~i~An~~~i$piele).
Habihtation thesis, Tech&a1 University of Berlin (1983).
16. 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 Lafayette, 23-25 May (1983).
17. D. Karamanlidis and S. N. Atluri, Mixed finite element
models for plate bending analysis: theory. Paper submitted for publication (May 1983).
18. 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.
19. 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).
20. F. Kikuchi On a mixed method related to the discrete
Kirchhoff assumption. Hybrid and Mixed Finite Element Models (Edited by S. N. Atluri et al.). Wiley, New
York (1983).
21. K. Knothe, Plattenberechnung nach dem Kraftgriissenverfahren. Der Stah~b~u 36. 202-214 and 234-254
f1%7j.
22. H. Le The, ~~nung
diinnez Schalen mit Hilb e&s
aemischt-hvbriden Fi~te-~~~t-M~al~s.
DimnIoma
Thesis (unpublished), Technical University of ‘Berlin
(1980).
23. R. J. Roark and W. C. Young, Formulasfor 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 Analysisof PlatesfClassictdattd
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
problems. Adurmces Engng Ssftwure 1, 137-140 (1979).
Mixed finite element models for plate bending analysis
28. J. P. Wolf, Generalized stress models for finite element
analysis. Report No. 77-ETH, Ziirich (1977).
29. J. P. Wolf, Das Fllchentragwerksprogramm
von
STRIP. Schweizerische Bauzeitung 90,41-52 (1972).
30. W. Wunderlich, Mixed models for plates and shells:
minciples-elements-examnles,
Hybrid and Mixed Pikite .t?lement Methods (Edited by S. N. Atluri et al.).
Wiley, New York (1983).
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]
581
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 Aycan 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.
&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 counterpart BDKT element (20) with nine DOF.
1
-lb.w.dA+~“~~~.(w,,+e~)
dA
I
+ 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 AXand A,.