A review and catalogue of plate bending finite elements

Compurers& Srmcmres Vol.19, No. 3, pp. 419-495, 1984 F’rintedin the U.S.A.

OlMs-7949/84 s3.00 + .oo Pergamon PressLtd.

A REVIEW AND CATALOGUE OF PLATE BENDING FINITE ELEMENTS M. M. HRABOK Cambrian Engineering, 119-105 Street East Saskatoon, Saskatchewan, Canada S7N 122 and T. M. HRUDEY Department of Civil Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G7 (Received 20 June 1983; received for publication 21 November 1983) Abstract-A brief review of developments in the field of plate finite elements is presented. This review is followed by an extensive tabular listing of plate bending elements.

lNTRODUCI?ON The finite element method has undergone an extremely active development period since its beginnings in the late 1950s. During the first ten to flfteen years of this activity, research effort centred on element development. More recently there has been a natural trend towards work focussing on applications of the method. One of the lirst tasks facing a potential practitioner of the finite element method is element selection. At this stage, one is confronted with the bewildering array of elements that has resulted from over twenty years of research activity. An area for which this problem is particularly difficult is that of plate bending where the number of available elements is large and no one particular element has emerged as the so called “best” element. The objective of this paper is to report the results of an extensive review of research on finite elements for thin plates. This review was conducted as part of a project to develop a finite element program for analysis of stiffened plates as encountered in reinforced concrete floor systems[l]. The results of this review are presented here in a form it is believed will be particularly useful. The main body of the paper consists of two parts. The first is an historical review of developments in the field of plate bending elements. It is deliberately brief and is intended to serve as background for the main contribution which is a tabular presentation of existing plate elements. The geometry and nodal configuration of each element is shown along with other pertinent details. This table or catalogue will serve as a useful guide and reference for researchers, educators and prospective element users. Coupled with the review and table is an extensive bibliography. Developments in plate bending jinite elements Interest in plate bending elements came very early in the history of the finite element method. At the beginning of the 1960s a number of elements were proposed by researchers such as Clough [2], Adini [3], Melosh[4] and Tocher[S]. These elements, as with most others developed in that period, were of the displacement type. By the mid 1960s the variational basis for the finite element method had become better understood, and 479

coupled with this, came the realization that interelement compatibility or conformity was an important property. Without it element convergence might not always be obtained. The conformity requirement for plate problems proved to be particularly problematic in that interelement continuity is required for both the transverse displacement and the slope normal to the element boundary. Most early plate bending elements were of the nonconforming type. Success in achieving full conformity came easiest in the case of rectangular elements. Bogner et al.[6] developed 16 and 36 degree of freedom conforming rectangles that exhibited good convergence properties. It was necessary, however, to use second derivatives of displacement as degrees of freedom. In particular, for the 16 degree of freedom element the twist was used. It was explained later by Irons and Draper[q that it is not possible to derive a conforming element using simple polynomials and only the three geometric degrees of freedom. The derivation of suitable triangular elements proved to be considerably more difficult than rectangular elements. Clough and Tocher [2] discussed three of the early non-conforming triangular elements. One of these is an element that was derived earlier by Adini[8], and which violates the constant strain requirement and does not converge. The second, derived by Tocher [5], converges but is not geometrically isotropic and cannot be derived for certain shapes. The third does not pass the patch test and was found to converge to incorrect results. Bazely et al.[9] simplified the problem associated with geometric isotropy by formulating shape functions using area coordinates. Although their nonconforming triangle has been used widely, it does not satisfy the patch test. Satisfying the conformity requirement was found to be particularly difficult for triangular elements. A number of different approaches were followed. To achieve a conforming element, while still using only the three geometric degrees of freedom, Bazeley et al. introduced the approach of superimposing non-polynomial shape functions. Zienkiewicz[lO] has labelled these types of displacement functions as “conforming shape functions with nodal singularities”. This term refers to the fact that the second

480

M. M. HRABOKand T. M. HRUDEY

derivatives or curvatures are not defined uniquely at the element nodes. Clough[2] introduced what Gallagher[ 1l] has labelled as the “subdomain approach” in which the main triangle is subdivided into three component triangles. The matrix for the whole triangle is obtained by imposing compatibility conditions along subtriangle boundaries. Several researchers have used the subdomain approach or variations of it. Elements derived in this way are identified in Table 2 by showing the subtriangles within the element. Of all the different schemes considered for the derivation of conforming plate elements, perhaps the most straightforward is the use of higher order polynomials. This class has been referred to as refined or higher order elements. A twenty-one degree of freedom element having mid-side nodes and using a complete quintic polynomial displacement function was developed simultaneously by Argyris [ 121, Bell [ 131, Irons [ 141 and Visser [ 151. Constraining the variation of normal slope to be cubic along the element boundary eliminates the need for the three mid-side nodes. The resulting eighteen degree of freedom element was developed simultaneously also by three different groups: Argyris[l2], Be11[13] and Cowper et al.[16]. Conforming plate elements were not only difficult to obtain, but with the exception of the higher order elements, they were found to be too stiff. There was considerable scepticism about the need to meet the C’ continuity requirement and many researchers looked for alternate formulations. The subsequent research followed several different paths. A significant improvement to Bazeley’s approach was presented by Irons and Razzaque[l7] through the use of “substitute shape functions” and the “smoothed derivative technique”. The substitute shape functions replace certain terms in the original functions. The degree of the highest complete polynomial is not changed and the derivatives in the energy functional are approximated in a least squares sense[lO]. The resulting elements are not conforming except in the limit as the element size is decreased. Other researchers experimented with a complete quadratic polynomial and obtained the “constant moment triangle”. The counterpart of this element in plane elasticity is the “constant strain triangle”. Still other researchers sought elements based on alternative variational principles. One logical choice is the principle of minimum complementary potential energy resulting in an “equilibrium formulation”. In principle it would appear that all that is required is to choose interpolation functions for the stresses or moments within the element. The chosen functions would be required to satisfy equilibrium at every point in the structure and the stress conditions on the boundaries. However, as described by Zienkiewicz[lO], “despite many trials of horrifying complexity” seldom has this been achieved directly with stresses as variables. One of the major difficulties arises in satisfying the kinematic boundary conditions. Initial work in this field was done by de Veubeke [ 181.To avoid a redundant force analysis, de Veubeke formed element flexibility matrices directly, inverted them to get stiffness matrices and then proceeded with a displacment type of solution. Prob-

lems arose with this approach when the assembled stiffness matrix was found to be positive semidefinite, indicating a kinematically unstable structure. Another means of using the complementary energy principle is to use the “flexibility” or “force” approach where a set of redundant self-equilibrating forces is chosen as the unknowns. In finite element analysis, difficulties with automating the selection of the redundant force system have caused this approach to be all but abandoned. Considerable clarification and simplification in the use of the equilibrium method can be attributed to Morley[l9,20] and Elias[21], who implemented the use of element stress functions. Stresses are calculated from the second derivatives of these functions and therefore the stress functions must still possess C’ continuity. However choosing these functions is made easier by “the principle of duality”. These analogies have been discussed extensively in the literature by Southwell[22], de Veubeke and Zienkiewicz [23], Morley[l9], Elias [21] and Sander[24]. In spite of the contributions of Morley and Elias, difficulties still exist with choosing the stress functions, defining the applied load state and specifying the boundary conditions for the stress functions. As well, the displacements do not possess unique values because they are obtained from integrating the strains. More detailed discussions on obtaining solutions using the principle of minimum complementary energy are given in the texts by Gallagher [25] and by Zienkiewicz [ lo]. Some other formulations that followed made use of Lagangian multipliers to allow relaxation of the continuity requirements along interelement boundaries thereby reducing the conformity requirement to C? continuity. The use of these modified functionals led to the hybrid methods and the generalised mixed methods. The term “hybrid” is used here to refer to formulations where one set of unknowns can be eliminated at the element level. The term “generalised mixed methods” refers to formulations where both sets of unknowns appear in the global set of equations. The tirst hybrid method developed was the stress hybrid method of Pian[26,271. Using a modified complementary potential energy principle, Pian chose stress polynomials for the interior of the element and displacements around the perimeter of the element. The latter play the role of Lagrangian multipliers to force interelement equilibrium. This method is known as the hybrid stress method. Other researchers, such as Tong[28] and Kikuchi and Ando[29], developed various displacement hybrid approaches based on modified forms of the principle of minimum potential energy. Another alternative to the displacement method was presented by Herrmann[30] in 1965. It is a mixed method and is based on a modified Reissner variational principle. Using this method, different combinations of displacements and stresses can be assumed on the interior as well as on the boundaries of the element. Herrmann relaxed the continuity requirements for displacements but imposed continuity conditions on the stress field. The result was that Co continuity was required of both sets of trial functions. Computations for the element stiffness are reduced because lower order polynomials can be used, but

R I D

H V

i

and

and

Continuous Equlllbratlng Stresses

Modlf led Complementary Energy

EGUILIBRIUN

and

Continuous Stre5s and Disolacement Fun& Ions Continuous Displacements

I

Continuous Displacements

Continuous Displacements

Modif led Potential Energy

Relssner Method a5 modified bv Herrmann _

I

Continuous Equillbratlng Stresses

Continuous Equlllbrating Stresses

DISPLACEMENT

I

Modif led Potent la1 Energy

HYBRID DISPLACEMENT METHOD(P)

REISSNER’s PRINCIPLE

Wodlf led Potential Energy

I

Modlffed Complementary Energy

Mlnlmum Complementar Energy

HYBRID DISPLACEMENT METHOD( 11

I

HYBRID STRESS METHOD

EPUILIBRIUM

Functions the element

Continuous Displacements

inside

Assumed

I

I

Lagranglan multlpllers (dlsplacements1

Lagranglan multipliers (stresses)

Combinations of Boundarv Tractions and Displacements

Assumed Equilibrating Boundary Tractlons

Assumed Compatible Displacements

Assumed Compatlble Displacements

Equlllbrium Boundary Tract ions

Dlsplacement Compatlbllity

Along Inter-element Boundaries

I

I

Noda 1 Displacements

Generalized Displacements Stress Parameters

Nodal Displacements and Lagranglan multipliers

Nodal Displacements and Lagranglan multlpllers

Combinations of Dlsolacements and’ Tract Ions

displacements and Boundary Forces

Nods 1 Displacements

Nodal

b)

a)

Noda 1 Displacements

Unknown5 in Final Equations

Table 1. Classification of finite element methods

I

I

a)

de

Veubeke[64]

P1an[261,[651

Worley~19l.[201 Ellas[21]

Anderheggen(351

Greene et a11321 Anderheggen(33) Harvey6 Kelsey[34]

Herrmann[30],[661 Pian 6 Tow1271 _-

GallagherI25)

Tong[26] Kikuchi & Ando[29]

bl

Courant[62] Melosh[63]

REFERENCES

482

M. M.

HRABOK

and T. M. HRUDEY

Table 2. Existing plate elements ELEMENT 12 dof

1)

P

n 12 dof

2)

ACM

I 12 dof

3)

M

I 4)

8)

24 dof

Displacement Type (Conforming) - Kirchhoff Plate Theory - Nodal dof=< w w,. w,~ w,.. w,.y w.*)r ’ - References: Bogner et a1[6]Hermitian functions, Popplewell and MacDonald[761, Gopalachar ulu [771- quartic polynomial. Watkins178 v , 1791 - blended Hermitians,

36 dof

Displacement Type (Conforming) - Kirchhoff Plate Theory - Nodal dof= < w w,. w,~ w...

n n

9)

- References:

Bogner,Fox

W,..Y wrxlv* Wf.YY w..*

w.*)r

’

and Schmit[Cl.

12 dof

Displacement Type (Nonconforming) - Kirchhoff Plate Theory - Nodal dof= < w w,~ w,~ > - References: Dawe[80] - modified the ACM polynomial to reduce the coefficients which were causing the wrn discontinuity.

12 dof

Mixed TypetGeneralized Displacement Method1 - Kirchhoff Plate Theory - Nodal dof= < w w,. w,v ’ - References: Greene,Jones,McLa and Stromet811, Harvey and Kelsey v 341; (triangles). - Lagrangian multipliers are used at a global level to restore continuity.

16 dof

Hybrid Stress Type - Kirchhoff Plate Theory - Nodal dof= < w w,. w,Y wrlv ’ - References: Pian and Tong[821, Piant - spurious energy modes may appear for thir and other elements if assumed moments arc linear - Pian & Mau1841, HolandI851.

i 10)

with a side.

Displacement Type (Conforming) - Kirchhoff Plate Theory - Nodal dof- < w w,. w,.y ’ w,* - References.: Bogner,Fox and Schmit[Cl Butlin and Leckie[731, Hansteen[741, Mason[751.

IX 7)

Displacement Type (Nonconforming) - Kirchhoff Plate Theory - Nodal dof= < w w,~ w.y > - References: Melosh[L] - cubic beam functions along edges linear variation to the opposite

16 dof

n

term

Displacement Type (Nonconforming) - Kirchhoff Plate Theory -Nodaldof=
Hybrid Stress Type - Kirchhoff Plate Theory - Nodal dof= < w wrx w.y ‘0 - References: Pian -also stress-free edges, Severn and Taylor[701, Henshell and co-workers[71,72] - various combinations of w and M .

BFS-16

6)

Displacement Type (Conforming) - Kirchhoff Plate Theory - Nodaldof=
12 dof

n 5)

DESCRIPTION

n

A review and catalogue of plate bending finite elements Table 2. (Contd) DESCRIPTION

EL~NT 1)

36 dof

Displacement Type (Conforming) - Love’s Moderately Thick Plate Theory - Nodal dof= < w wtx w,,. W,XX wrrr w,*y ’ w*.xv wIxIY w,x.,y - References: Smith[86], Smith and Duncant - also form a 24 dof rectangle by ignoring the last 3 nodal dof.

24 dof

Displacement Type (Ronconforming) - Kirchhoff Plate Theory - Nodal dof=< w w,. w,y w,.. w,x* w,ry ’ - References: Wegmuller and Kostem[883 WegmullerC89,901 - complete quintic polynomial and the terms x’y’, and xyl. r’y,

12

Hybrid Displacement TypeiSimplified Method: - Kirchhoff Plate Theory - Nodal dof= < w If,* w.” ’ - References: Kikuchi and Andof - derived 4 rectangles and 4 triangles by using.various displacement combinations, <‘corrective’ matrix enforces continuity’

n 2)

i 131

dof

I 14)

8 dof CMR

n 1

2 16 dof

15)

~ 1

2 16 dof

161

r% 1

2

16 dof 17) Semi-Loof

1 2 18)

12 dof Bi .MPT

II

19)

24 dof QSR

pi 20)

27 dof

QLR

Hybrid Displacement Type - Kirchhoff Plate Theory -Nodeldof=< w > or 34. zNode2 dof= < w,. - References: Kikuchi and Ando[291: (as above) Poceski[Sll -mixed method, - this is the ‘Constant Woment Rectangle’. Displacement Type (Discrete Kitchhofff - Mindlin Plate Theory - Node1 dof= < w 0. 6” > Node2 dof= < - References: fa1si.a p&lelogramf Razzaque[92j, Baldwin[49F, Irond511. - reduced 24 dof to 16 by using 8 discrete conditions at the Gauss points., Displacement Type (Discrete Kitchhofff - Mindlin Plate Theory - Node1 dof= < w 6. 8, > Node2 dof= < - References: (a1st.a qua&ilateral) Baldwin,Razzaque and Irons[49] - reduced 25 dof to 16 by using constraint at 8 Loof nodes and s perimeter integral Displacement Type (Discrete Kirchhoff) - Mindlin Plate Theory - Bode1 dof= < w > Node2 dof- -z 8, z , (Loof nodes) - References: Irons[511, Martins and Owens[931 - reduced 27 dof to 16 by using constraint at 8 Loof nodes and 3 area integrals. Displacement Type (Selective Integration) - Mindlin Plate Theory - Nodal dof- ( W 8. 0, > References: Pughl941, Pugh et al I44l, Hughes,Taylor and Kanoknukulchai[Qlf - bilinear displacement functions, - two spurious energy modes. Displacement Type (Selective Integration) - Mindlin Plate Theory - Nodal dof- < w 8. e, > -.References:(Quadratic Serendipity Reduce1 PughL941, Pugh et el 1441 - basically the same as Ahmad’s reduced integration plate element; may diverge or converge.erratically [44]. Displacement Type (Selective Integration1 - Windlin Plate Theory - Nodal dof- < w 8. e, ? - References: (Quadratic La range Reduced) Pugh[941, Pugh et al [44 s -- four spurious energy modes for S2x2 reduced integration, z can be mappad into a quadrilateral.

483

M. M. HRABOKand T. M. HRULW Table 2. (Cod)

2)

48 dof

reduced

integration.

- Nodal dof= - References:

< v

nodes and 3 shear

Node2 dof= - Ref erencts:

<

- Node2 dof=

<

integrals.

lso

44 and 66 dof

- Kirchhoff Plate Theory - Nodal dof= < w w,x w,.y ’ - References: AdiniK81, Clough and Tocher[Zl - constant twist term ‘xy’ omitted: erroneous convergence (too stiff).

- Kirchhoff Plate - Nodal dof= < w - References:

Theory w,. w,y ’

- Node1 dof= Node2 dof= - References:

Y

< Y <

elments)

A review and catalogue of plate bending finite elements Table 2. (Conki) ELEMENT

DESCRIPTION

\;_-D

11)

9 dof HCT

/’

9 dof B C I Z(nc)

32)

P 9 dof 33) B C I Z(c)

P 34)

12

dof

P 9 dof

35)

D 9 dof

36)

P 9 dof

37)

I\. 38)

21

T-21

dof

Q 2

1

39)

CUT

D 1 40)

6 dof

2 12 dof

Displacement Type (Conforming) - Kirchhoff Plate Theory - Nodal dof- < w w,. - References: (Hsieh-C1~~~hzTocher element) Clough and Tocher[Z] - beginning of the ‘subdomain approach’, - ‘geometric isotropy’ preserved by special choice of axes; linear w,” enforced. Displacement Type (Nonconforming) - Kirchhoff Plate Theory - Nodal dof- < w w,. w#I ’ - References: Bazeley,Cheung,Irons and Zienkiewicz191 - introduced the use of ‘area coordinates’ to retain geometric isotropy: also begin shape function’ approach. the ‘substitute Displacement Type (Conforming) - Kirchhoff Plate Theory - Nodal dof= < w w,. w,), ’ - References: Bazeley,Cheung,Irons and Zienkiewiczlgl - as above, but corrective shape functions used to obtain conformity (very stiff), - requires very high order of integration. Mixed Type (modified Reissner - Reissner Plate Theory - Nodal dof= < w M. M, W., - References: Herrmann[30] Chattcrjee and Setlurr971 - assumed linear variation of - trial functions need only C’

Principle) ’

w and M continuit;.

Displacement Type (Discrete Kirchhoff) - Mindlin Theory or (Thick Plate Theory displacement functions for u, v, and w). - References: Melosh[9S], Utku[991,[1001, Martin[lOl Wempner et a1[471, Dhatt[102,1031, Stricklin et a1[48], Fried[50,104], Hinton et al[lOS], Batoz et al [106]. Equilibrium Type(Argyris’ Natural Approach1 - Kirchhoff Plate Theory - Nodal dof- < w w,. w,y ’ - References: Argyris[l07] - obtained a 6x6 flexibility matrix by using the Unit Load method. Hybrid Stress Type - Kirchhoff Plate Theorv - Nodal dof= -z w w,. ;.y ’ - References: Severn and Taylort701 - assumed quadratic M and cubic

w

.

Displacement Type (Conforming) - Kirchhoff Plate Theory - Node1 dof=< w w,. wP,* w... w,x* wry* ’ - Node2 dof= < w,. - References: Felip [:OS] Withum[lOSl Argyris[l2], Bell p”1101, Bosshard[llll Visser[llZl, Irons11131 - used a complete quintic polynomial. Methods : CONSTANTMOMENTTRIANGLE Displacement ; Horley[ll41 Eouilibrium : Allman[ll51 Hybrid Stress: Yoshida[ll61 Hybrid Disp. - Kikuchi and Andor Mixed iHerrmannr661, Hellan[ll71 - Node1 dof= < w or Q> Node2 dof- < w,* Or M. > Equilibrium Type (Duality Approach) - Kirchhoff Plate Theory - Nodal dof- < er, 4r > - References: Morley[19,201, SanderL241 - quadratic moment functions, - nodal parameters are the Southwell stress functions.

485

M. M. HRABOK and T. M. HRUDEY

486

Table 2. (Co&) ELEMENT 9 dof

41)

p 12 dof Lcct-12

42)

\

,>___ D’

2

1

15 dof T-15

43)

D 1

2

9 dof CPT

44)

\ \\ p 45)

12 dof LI.lT

p 9 dof

46)

\

&_--

/’

p 47)

18 dof T-18

p 48)

21 dof TUBA-6

D 1 49)

2

28 dof TUBA-13

. p 1 50)

36 dof TUBA-15

. . B 1

DESCRIPTION Hybrid Stress Type - Kirchhoff Plate Theory - Nodal dof= < w w,. w,y ’ - References Dungar,Severn and Taylor[llB], Allman[llS], Neale,Henshell and Edwards[ll91, Yoshida[lZOl, Batoz, Bathe and Ho[1061. Displacement Type (Conforming) - Kirchhoff Plate Theory - Nodal dof= < w w,. W.” ’ > - Node2 dof- < w.. - References: Clough and Felippa[l21] -improved the subdomain approach by using complete cubic and reducing 30 dof to 12.

i

Displacement Type (Nonconforming) - Kirchhoff Plate Theory - Nodal dof- < w w,. w,y ’ > - Node2 dof= < w w,. - References: Be11[13,1101, Chu and Schnobrichfl221 - used complete quartic shape functions. Displacement Type (Non-Conforming in w) - Kirchhoff Plate Theory - Nodal dof= < w w,. W,I ’ - References: Connor and Wi11[123] - discarded the x’y term, conforming in w,, but not in w along one of the sides, - used in the STRUDLZ computer program. Equilibrium Type - Kirchhoff Plate Theory - Nodal dof= < w > - Side dof- < averaged integral value of w and 2 weighted edge rotations > - References: (Linear Moment Triangle) de Veubeke and Sander11241, Somervaille[lES] -also presented Q M T, Displacement Type (Conforming) - Kirchhoff Plate Theory - Nodal dof- < w w,. w,* ’ - References: Shieh, Lee, and Parmelee[l261 - used a quadratic function and reduced element cannot satisfy 18 dof to 9 dof, interior displacement comptability. Displacement Type (Conforming) - Kirchhoff Plate Theory - Nodal dof=< w wrx wev w,.. we.” w.y* - References: Cowper et a1[16,127], Argyris[lZl, Butlin and Ford[128], Be11[13,1101. - derived from T-21 triangle by imposing a cubic variation of w.. .

’

a

Displacement Type (Conforming) - Kirchhoff Plate Theory - Node1 dof= < w w,. w,* w,.. w,x). w,YY ’ Node2 dof= < w,. ’ - References: Argyris[lZ]identical to T-21, -complete quintic displacement polynomial, element has 6 nodes and 21 dof. Displacement Type (Conforming) - Kirchhoff Plate Theory - Node1 dof= < w w,. w,), w,.. w,.r w,** ’ dof at remaining nodet Nodei dof- various - References: Argyrisr121 -complete sextic displacement polynomial, element has 13 nodes and 28 dof. Displacement Type (Conforming) - Kirchhoff Plate Theory - Node1 dof- < w w,. w,” w... W,X” wtv)r ’ dof at remaining nodet Nodei dof= various - References: ArgyrisLl21, Zenisek11291 -complete septic displacement polynomial, element has 15 nodes and 36 dof.

487

A review and catalogue of plate bending finite elements Table 2. (Con@ ELEMBNT 51)

6 dof

52)

12 dof

53)

18

dof

D 2

54)

24 dof

P

DESCRIPTION Equilibrium Type - Kirchhoff Plate Theory - Nodal dof= < @., 0, > - References: Elias[Zl], Sanderr - nodal parameters are Southwell - assumed linear moment functions.

functions

Mixed Type (Herrmann’s Method) - Reissner Plate Theory - Nodal dof= < w M. M, M.” ’ - References: Visser[lSl, BootIl301 - 9 dof obtainable by static condensation, - parabolic variation of w and a linear variation of M . Displacement Type (Conforming) - Kirchhoff Plate Theorv - Node1 dof= < w w,. G,, > Node2 dof= < w w,. w,.t ’ - References: Irons1141 - used all 15 terms of a quartic plus functions. basic or ‘singularity’

3

Mixed Type(Generalised Equilibrium Method) - Linear Stress Variation across thickness - Nodal dof= i M. M, M., ’ - Side dof= < 2 Lagrange multipliers to restore shear continuity > - ‘References: Anderheggenf351, Meek[l311

55)

9 dof

Hybrid Stress Type - Kirchhoff Plate Theory - Nodal dof= < w .w,. w,* ’ - References: Dungar and Severn[l32], - various combinations of M and w, - variable thickness, also triangles with stress-free edges, and hybrid beams.

56)

9 dof

Hybrid Displacement Type - Kirchhoff Plate Theory - Nodal dof- < w w,, “,* ’ - References: Hansteen[l33], Yoshida[l161, Allman[1341 - Allman’s-cubic w for interior and edges is identical to a stress hybrid with M . cubic w and linear

57)

12 dof

Mixed Type(Generalized Displacement Method) - Kirchhoff Plate Theory - Nodal dof= < w w,. w,* ’ - Side dof= < weighted average of M. fused to restore interelement continuity)> - References: Anderheggen1331 one dof is integral of w - complete cubic,

58)

12 dof

Hybrid Stress Type - Kirchhoff Plate Theory - Node1 dof= < w wri “I” ’ > Node2 dof= < w,. - References: Allman[ll5], Bartholomew[1351 - element with linear M and cubic identical to Razxaque’s ‘A-12’.

P 2

59)

dof

9,lO

D 02

60)

12

dof

P

w is

Mixed Tgpe(Generalized Displacement Method) - Kirchhoff Plate Theory - Node1 dof= < w w,. w,r ’ - Node2 dof= < w >, can be condensed out. - References: Harvey and Kelsdy[34], Meek[l311 - Lagrangian multipliers restore continuity at a global level, similar to Anderheggen Mixed Type (Herrmann’s Method) - Reissner Plate Theory - Nodal dof= < w M. M, M,, > - References: Tahiani[l36], Bran and Dhatt[ 1371

M. M. HRABOKand T. M. HRUDEY

488

Table 2. (Co&) DESCRIPTION

ELEMENT 61)

24 dof

Mixed Type (Herrmann’s Method) - Reissner Plate Theory - Nodal dof= < w M, M, M., - References: TahianiLl361, Bron and DhattI1371

’

Displacement Type (Nonconforming) - Kirchhoff Plate Theory - Nodal dof= < w w,, w.), ’ - References: Irons and Raziaque[l7,92,138,1391 - used ‘derivative smoothing’ and ‘substitute shape functions’, - identical results to Allman[ll51 hybrid. 12 dof

63)

p 1

2

Displacement Type (Conforming) - Kirchhoff Plate Theory - Node1 dof= < w w,. w.y ’ > Node2 dof= < w,. - References: Irons and Razzaque[17,92,138,1391 - used ‘derivative smoothin ’ but element identical to Allman’s[ll5 hybrid s stress Hybrid Displacement Type(Simplified Method) - Kirchhoff Plate Theory - Nodal dof= < w w,. w,r ’ - References: Kikuchi and And01291 - used a complete cubic and a ‘corrective matrix’ to derive 4 rectangles and 4 triangles. Displacement

Type

(Conforming)

- similar to TUBA-15 but reduced to 33 dof, - 10 node element used for contact problems 66)

9 dof

Hybrid Stress Type - Love’s Plate Theory - Nodal dof= < w w,. w,* ’ - References: Cook[141,1421 - various aspects of the hybrid stress method; emphasis on transverse shear, - also formed quadrilaterals from triangles

67)

9 dof

‘Direct Approach’ - Kirchhoff Plate Theory - Nodal dof= < w w.. w,* ’ - References: Bergan and HanssenL561 - not based on any variational principle but must satisfy constant strain states and pass the ‘patch test’.

68) 2

8 dof

Mixed Type (modified Herrmann’s method) - Kirchhoff Plate Theory - Node1 dof= < w M. H; ‘, Node2 dof= < w M. M along 2-21 Node3 dof= < w > ,(lineZr (constant M along l-2) - References: Poceski[911 - stress polys. partly dependent on disps.

D 1

70)

2

9 dof

Displacement Type (Conforming) - Kirchhoff Plate Theory - Node1 dof- < w w,x w.y ’ Node2 dof= < w w.. w., w.., ’ - References: Caramanlian, Selby and Wi11[1431

Displacement Type (Selective Integration) - Mindlin Plate Theory - Nodal dof= < w 8. 8, > - References: Batos,Bathe and Ho[1061 - a study of 9 dof triangles which included hybrid stress, and conventional disp., discrete Kirchhoff elements.

489

A review and catalogue of plate bending finite elements Table 2. (Contd) DESCRIPTION

ELEMENT 711 12 dof Rhombics

0 72)

12 dof

Displacement Type(Argyris’ Natural Method) - Kirchhoff Plate Theory - Nodal dofw < w w,. w,y ’ - References: Argyris[l071 - derived 9x9 ‘natural flexibility matrix’ from which the 12x12 stiffness matrix was obtained.

12 dof

Displacement Type (Nonconforming) - Kirchhoff Plate Theory - Nodal dof= < w v,. w,* ’ - References: Dawe[ 1451, Ramstad and Holand[l461, Ramstad[lQIl - Dawe utts the ACM polynomial in an oblique coordinate system.

16 dof

Displacement Type (Conforming) - Kirchhoff Plate Theory - Nodal dof- < w w,. WI* w,.* - References: Granheim[l48]

D 73)

/T7 74)

Methods: Displacement ; SandtrL241 Equilibrium ; Sander1241 Hybrid Stress; Wolf[l44] - Nodal dof= < w wII w,* ’ - References: Pian[831

’

/7 75)

12,24,36 dof

0 16 dof

76)

CQ-16 cc 1 q 77)

q 78)

\

2 12 dof *.. *s ---\ \ \

24 dof . . , _-- y\

q 79)

Displacement Type (Conforming) - Kirchhoff Plate Theory - Node1 dof= < w w,. w,), ’

Node2 dof= < - References:

A./\

Equilibrium Type (Duality Approach) - Kirchhoff Plate Theory - Nodal dof- < 9., aY > - References: Sander1241 - derived a family of equilibrium linear, quadratic, and cubic paralltlograms and triangles (also sub- and hyptr- elements

\ \

16 dof

q SO) Polygons

-

w,.

>

Sanderrl491, de Veubeke[l8,1501 12 dof quad. may be obtained by imposing a linear variation of normal slopes.

Displacement Type (Conforming) - Kirchhoff Plate Theory - Nodal dof= < w wrr w,* ’ - References: Clough and Felippa[lZl] - the quadrilateral used in the SAP4 computer program. Displacement Type (Conforming) - Kirchhoff Plate Theory - Nodal dof=< w w,# w,” w,.. - References: Clough and Ftlippa[lZl]

w,.),

wIY* ’

Mixed TypetGeneralised Displacement Method - Kirchhoff Plate Theory - Nodal dof= < w w,. wSry > - References: Gretne,Jonts,McLay and Strome[32,811 - used Lagrangian multipliers to restore continuity at a global level. Hybrid Stress Type - Kirchhoff Plate Theory - Nodal dof= c w w,. w,y ’ - References: Allwood and Cornts[l511 - reported results for work done on polygons with 3 to 9 sides.

490

M. M. HRABOKand T. M. HRUDEY

Table 2. (CO&) I

ELEMENT

DESCRIPTION Displacement Type ‘(C” Conformity) - Ahmad’s quadratic thick shell element. - Nodal dof= < w 0. 0, > - References: Ahmad,Irons and Zienkiewicz[40,1521 - a plate element obtained from a degenerated solid, found to be too stiff for thin plates. Displacement Type (Selective Integration) - Ahmad’s shell element underintegrated. - Nodal dof- < w 6. 6” > - References: Pawsey and Clough[39] Zienkiewicz,Taylor and Too[38] - both ‘selective’ and ‘uniform’ reduced integration used to soften the element.

83)

12,24,36 dof

84)

16 dof

Mixed Type (Herrmann’s Method) - Reissner Plate Theory - Nodal dof- < w M. M, I&Y ’ - References: Bran and DhattL1371 - linear w and linear M.

85)

24 dof

Mixed Type (Herrmann’s Method) - Reissner Plate Theory - Nodal dof= < w M. M, M., ’ - References: Bron and Dhatt[1371 - quadratic w and quadratic M.

86)

12 dof

Hybrid Stress Type - Kirchhoff Plate Theory - Nodal dof= < w w,. v,y - References: Torbe and Church[l53] - also derived the in-plane

87)

24 dof

Mapping of Rectangles to Quadrilaterals. - Kirchhoff Plate Theory - Nodal dof= < Y w,. v,” ’ - References: Henshell.Walters and Warburton - a quadrilateral can be obtained from a rectangle by a transformation of coordinates, but the constant curvature states may be destroyed.

’ matrices.

Hybrid Displacement - Trefftz’s Principle - Kirchhoff Plate Theory - Nodal dof=< w *,. v.y v,., W,.” u,** > - References: Jirousek and Leon[1541 - independent disp. functions for interior and perimeter; attempt to satisfy the differential equations of equilibrium. Hybrid Stress Type (Selective Integration) - Mindlin Plate Theory - Nodal dof= < w 8. 6, > - References: SpilkerL541 - derived a series of 4-node quadrilaterals - also derived Serendipity quadratic and cubic elements for thick plates.

both equilibrium and compatibility are satisfied only approximately [1I]. In the “generalised displacement method”, nonconforming elements are used in conjunction with interelement Lagrangian multipliers. The Lagrangian multipliers are present in the global equations and can be identified as forces. This approach appears to

have been initiated by Jones[31], Greene et al.[32], Anderheggen[33] and Harvey and Kelsey[34]. Similarly, in the “generalised equilibrium method”, the Lagrangian multipliers are applied to the global set of equations and are identified as displacements. Such a solution was used in 1969 by Anderheggen[35] and later by Sander[24]. With these two mixed meth-

A review and catalogue of plate bending finite elements

ods, as with Herrmann’s formulation, Lagrangian multipliers are included in the final equations. These equations are positive semi-definite and therefore special care must be taken during the solution phase[25, lo]. An overview of the different element categories based on their variational formulation is presented in Table 1. This table is essentially the same as that published in 1969 by Pian and Tong[27] except for the addition of the generalised displacement and the generalised equilibrium methods to the mixed category. The decade of the 1970s saw a reduction in the rate at which new elements were derived. Nevertheless, research continued, resulting in the introduction of techniques such as reduced integraton and penalty number formulations, substitute shape functions and derivative smoothing, and discrete Kirchhoff constraints. One of the most significant developments which emerged from this decade was the use of the displacement formulation based on Mindlin plate theory and reduced integration schemes. The motivation for using Mindlin plate theory is that only first derivatives appear in the energy functional and therefore only Co continuity is required of the shape functions. As well, the shape functions from plane elasticity elements can be used for the plate bending and the elements can be mapped isoparametrically. This approach works well for thick plates. However as the plate thickness is decreased, the shear terms dominate in the stiffness matrix and the socalled “locking” phenomenon occurs. In the limit as the thickness decreases, the deflections go to zero [36]. In this case it becomes necessary to either impose the Rirchhoff normality conditions directly as constraints or else make the strain matrix rank deficient by using reduced integration. The latter had been used with some success in 1969 by Doherty et al. [37] for overly stiff plane stress elements. For plate bending, it was introduced simultaneously in 1971 by Zienkiewicz et al. [38] and by Pawsey and Clough[39]. In both publications, its use was demonstrated on Ahmad’s eight-node serendipity shell element[40]. In evaluating the shear strain energy, both research groups purposely underintegrated the shear strain terms while intergrating the flexural terms exactly. This procedure is now known as “selective reduced integration”. Zienkiewicz et al. also tried underintegrating all terms by one order of integration. This is now referred to as “uniform reduced integration”. Both methods gave much improved results and led to the method of selective reduced integration with Mindlin plate theory. This was to dominate the plate bending research field from the mid 1970s to the present. The main contributions to this area have come from research groups associated with Hughes[41-43] and Hinton[44,45]. One alternative to using selective reduced integration for Mindlin plate elements is to impose, at the element level, the Kirchhoff constraints of normality at discrete locations such as the Gauss integration points or the Loof nodes[46]. This idea was introduced in 1968 by Wempner et al.[47] and has been used for plates by researchers such as Stricklin[48], Baldwin[49], Fried[50], Irons[Sl] and Lyons[52]. It is

491

not always successful and C’ continuity

is not always preserved[lO]. A second alternative is to impose the Kirchhoff constraints in a weighted integral sense by using Lagrangian multipliers that are interpolated over the element. This method has been tested by Hrabok[53]. Neither of the last two methods is as simple or effective as the selective reduced integration technique. Recently the use of the selective reduced integration scheme has been extended to hybrid stress formulations[54]. The many formulations discussed thus far are not all independent and a more comprehensive treatment of these topics and the equivalences between certain methods are discussed in the texts by Zienkiewicz [ lo] and by Gallagher[25]. As well, Malkus and Hughes[55] and Spiker[54] have shown the equivalence of selective reduced integration and some mixed methods. A radically different and promising approach is the direct method introduced in 1975 by Bergan and Hanssen[56]. This method is not based on a variational principle. Instead, a triangular element is derived directly from the conditions that it satisfy the patch test and the rigid body motion and constant strain requirements. The computations required are not as involved as those of some of the previous methods, and the resulting element has been found to be quite accurate. Also requiring mention is the so called “p convergent” or constraint method[57-61). With this method, refinement is achieved by increasing the order of the displacement polynomial within each element while the gridwork remains unchanged. The polynomial coefficients for the assembly of elements are constrained in order to satisfy nodal interelement compatibility. Table 2 shows all the elements discussed above as well as many others. In general the elements appear in chronological order with rectangles first followed by triangles and then quadrilaterals. Accompanying this table is an extensive bibliography. REFERENCES

1. M. M. Hrabok and T. M. Hrudey, Finite element analysis in design of floor systems. J.~Structural Engng, AXE 109 (3). . ,. 909-925 (1983). 2. R. W. Clough and J. L. Tocher; Finite element stiffness matrices for the analysis of plate bending. Proc. 1st Co@ on Matrix Meth. in Structural Mech., Ohio, pp. 515-545 (Oct. 1965). 3. A. Adini and R. W. Clough, Analysis of plate bending by the finite element method. Report submitted to the National Science Foundation (Grant G7337), Washington, D.C. (1960). 4. R. J. Melosh, A stiffness matrix for the analysis of thin plates in bending. J. Aeronaut. Sci. 28 (l), 34-42 (1961). 5. J. L. Tocher, Analysis of plate bending using triangular elements. Thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy, Department of Civil &ineering, University of California, Berkeley (1962). 6. F.-K. Bogner R.. L. Fox and L.‘A. Schmit, Jr, The generation of inter-element compatible stiffness and mass matrices by use of interpolation formulas. Proc. 1st Conf. on Matrix Methoak in Structural Mech., Ohio, pp. 397-443 (Oct. 1965).

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M. M. HRAB~Kand T. M. HRUDEY

7. B. M. Irons and K. J. Draper, Inadequacy of nodal connections in a stiffness solution for plate bending. Am. Inst. Aeronaut. Astronaut. J. 3 (5), 961 (1965). 8. A. Adini, Analysis of shell structures by the finite element method. Thesis presented in partial fulfilhnent of the requirements for the degree of Doctor of Philosophy, Department of Civil Engineering, University of California. Berkelev (1961). 9. G.-P. Bazeley, Y: K. Cheung B. M. Irons and 0. C. Zienkiewicz, Triangular elements in plate bendingconforming and non-conforming solutions. Proc. 1st on Matrix Meth. in Structural Mech., Ohio, pp. 547-576 (Oct. 1965). 10. 0. C. Zienkiewicz, The Finite Element Method in Engineering Science. McGraw-Hill, New York (1977). 11. R. H. Gallagher, Analysis of plate and shell structures. Proc. Symp. on Application of Finite Element Methods in Civil Engng, School of Engineering, Vanderbilt University, Nashville, Tennessee, pp. 155-205 (Nov. 1969). 12. J. H. Argyris, I. Fried and D. W. Scharpf, The TUBA family of plate elements for the matrix displacement method. The Aeronaut. J. Royal Aeronaut. Sot. 72, 701-709 (1968). 13. K. Bell, A refined triangular plate bending finite element. Int. J. Num. Meth. Engng 1 (1), 101-122 (1969). 14. B. M. Irons, A conforming quartic triangular element for plate bending. ht. J. Num. Meth. Engng 1 (I), 2945 (1969). 15. W. Visser, A refined mixed type plate bending element. Am. Inst. Aeronaut. and Astronaut. J. 7 (9), 1801-1803 (1969). 16. G. R. Cowper, E. Kosko G. M. Lindberg and M. D. Olson, Formulation of a new triangular plate bending element. Trans. Canadian Aeronaut. Space Inst. 1 (2), 86-90 (1968). 17. B. M. Irons and A. Razzaque, Shape function formulations for elements other than displacement models. Presented at the Symp. on Variational Meth. in Engng, University of Southampton, pp. 4/59-71 (1972). 18. B. Fraeijs de Veubeke, Bending and stretching of plates-special models for upper and lower bounds. Proc. 1st Co& on Matrix Meth. in Structural Mech., Ohio, pp. 863-886 (Oct. 1665). 19. L. S. D. Morley, A triangular equilibrium element with linearly varying bending moments for plate bending problems. The Aeronaut. J. Royal Aeronaut. Sot. 71, 715-719 (Oct. 1967). 20. L. S. D. Morley, The triangular equilibrium element in the solution of plate bending problems. The Aeronaut. Quart. 19, 149-169 (1968). 21. Z. M. Elias, Duality in finite element methods. J. Engng Mech. Div., ASCE 94, (EM4), pp. 931-946 (1968). 22. R. V. Southwell, On the analogues relating flexure and extension of flat plates. Quart. J. Math. Appl. Mech. III, Part 3, 257-270 (1950). 23. B. Fraeijs de Veubeke and 0. C. Zienkiewicz, Strain energy bounds in finite element analysis by slab analogv. J. Strain Anal. 2 (4), 265-271 (1967). 24. G-Sander, Applications of the dual analysis principle. Proc. IUTAM Svmv. on High Sueed Computing of Elastic Structures (Edited by-B. Fraeijs de Veubeke), pp. 2328, 167-207, University of Liege, Belgium (Aug. 1970). 25. 6. ?.. Gallagher, Finite Element Analysis: Fumiamentals. Prentice Hall. Enalewood Cliffs. New Jersev _ (1975). 26. T. H. H. Pian, Element stiffness matrices for boundary compatibility and for prescribed boundary stresses. Proc. 1st Conf. on Matrix Meth. in Structural Mech., Ohio, pp. 457-477 (Oct. 1965).

27. T. H. H. Pian and P. Tong, Basis of finite element methods for solid continua. Int. J. Numer. Meth. Engng 1 (1), 3-28 (1969). 28. P. Tong, New displacement hybrid finite element models for solid continua. Int. J. Num. Meth. Engng 2 (I), 7383 (1970). 29. F. Kinkuchi and Y. Ando, Some tinite element solutions for plate bending problems by simplified hybrid displacement method. Nuclear Engng Design 23, 155-178 (1972). 30. L. R. Herrmann, A bending analysis for plates. Proc. 1st Conf. on Matrix Meth. in Structural Mech., Ohio, pp. 577-602 (Oct. 1965). 31. R. E. Jones, A generalization of the direct-stiffness method of structural analysis. Am. Inst. Aeronaut. Astronaut. J. 2, (5), 821-862 (1964). 32. B. E. Greene, R. E. Jones, R. W. McLay and D. R. Strome, Generalized variational principles in the finite element method. Am. Inst. Aeronaut. Astronaut. J. 7 (7), 1254-1260 (1969). 33. E. Anderheggen, A conforming triangular finite element plate bending solution. Int. J. Numer. Meth. Engng 2 (2), 259-264 (1970). 34. J. W. Harvey and S. Kelsey, Triangular plate bending elements with enforced compatibility. Am. Inst. Aeronaut. Astronaut. J. 9 (6), 1023-1026 (1971). 35. E. Anderheggen, Finite element plate bending equilibrium analysis. J. Engng Mech. Div., ASCE 95 (EM4), 841-857 (1969), 36. 0. C. Zienkiewicz and E. Hinton, Reduced integration, function smoothing and nonconformity in finite element analysis. J. Franklin Inst. 302 (5,6), 443-461 (1976). 37. W. P. Doherty, E. L. Wilson and R. L. Taylor, Stress analysis of axisynnnetric structures using higher order quadrilateral elements. Structural Engineering Lab. Rep. SESM 69-3, University of California, Berkeley (1969). 38. 0. C. Zienkiewicz, R. L. Taylor and J. M. Too, Reduced integration technique in general analysis of plates and shells. Int. J. Num. Meth. Engng 3 (2), 275290 (1971). 39. S. F. Pawsey and R. W. Clough, Improved numerical integration of thick shell finite elements. Int. J. Num. Meth. Engng 3 (4), 575-586 (1971). 40. S. Ahmad, B. M. Irons and 0. C. Zienkiewicz, Curved thick shell and membrane elements with particular reference to axisymmetric problems. Proc. 2nd Conf. on Matrix Meth. in Structural Mech., Ohio, pp. 539-572 (1968). 41. T. J. R. Hughes, R. L. Taylor and W. Kanoknukulchai, A simple and efficient finite element for plate bending. Int. J. Num. Meth. Engng 11 (lo), 1529-1543 (1977). 42. T. J. R. Hughes and M. Cohen, The “Heterosis” finite element for plate bending. Comput. Structures 9, 445450 (1978). 43. T. J. R. Hughes, M. Cohen and M. Haroun, Reduced and selective integration techniques in the finite element analysis of plates. Nuclear Engng Design 46, 203-222 (1978). 44. E. D. L. Pugh, E. Hinton and 0. C. Zienkiewicz, A study of quadrilateral plate bending elements with “reduced integration”. Int. J. Num. Meth. Engng 12 (7), 10591079 (1978). 45. E. Hinton and N. Bicanic, A comparison of Lagrangian and serendipity Mindlin plate elements for free vibration analysis. Comput. Structures 10, 483493 (1979). 46. H. W. Loof, The economical computation of stiffness of large structural elements. Presented at The Symp. on the Use of Electronic Digital Computers in Structural Engng, University of Newcastle upon Tyne (1966).

A review and catalogue of plate bending finite elements 47. G. A. Wempner, J. T. Cklen and D. A. Dross, Finite element analysis of thin shells. J. Engng Me&. Diu. AXE 94. (EM6). 1273-1294 (1968). 48. J. A. Stricklin, W. E. Haisler,‘ P. R. Tisdale and R. Gunderson, A rapidly converging triangular plate element. Am. Inst. Aeronaut. Astronaut. J. 7 (I), 180-181 (1969). 49. J. T. Baldwin, A. Raxxaque and B. M. Irons, Shape function subroutines for an isoparametric thin plate element. ht. J. Numer Meth. Engng __ 7 (4). .,- 431-440 (1973). 50. I. Fried, Shear in C? and C’ bending 6nite elements. Znt. J. Solia!s Structures 9 (4). . _. 449-460 (1973). 51. B. M. Irons, The semi-loof element. Finite Elements for Thin Shell and Curved Members (Edited by G. H. Ashwell and G. H. Gallagher), Chap. 11. Wiley, New York (1976). 52. L. P. R. Lyons, A general finite element system with special reference to the analysis of cellular structures. Thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy, Department of Civil Engineering, Imperial College of Science Technology, London (1977). 53. M. M. Hrabok, Analysis of stiffened plates by the hybrid stress finite element method. Thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy, Department of Civil Engineering, University of Alberta, Edmonton, Canada (1981) 54. R. L. Spilker and N. I. Munir, The hybrid-stress model for thin plates, Int. J. Num. Meth. Engng 15 (8), 1239-1260 (1980). 55. D. S. Malkus and T. J. R. Hughes, Mixed finite element methods--reduced and selective integration techniques: a unification of concepts. Cornput-Meth. At&. Mech. Enmra 15, 63-81 (1078). 56. P: G. Bergan aid L. Hanssen; A new approach for deriving “good” element stiffness matrices, The Mathematics of Finite Elements and Applications (Edited by J. R. Whiteman), pp. 483497. Academic Press, London (1975). 57. B. A. Szabo and C.-T. Tsai, The quadratic programming approach to the finite element method. Int. J. Num. Meth. Engng 5, 375-381 (1973). 58. B. A. Sxabo and T. Kassos, Linear equality constraints in finite element approximation. Int. J. Num. Meth. Engng 9, 563-580 (1975). 59. M. P. Rossow, J. C. Lee and K. C. Chen, Computer implementation of the constraint method. Comput. Structures 6, 203-209 (1976). 60. M. P. Rossow and A. K. Ibrahimkhail, Constraint method analysis of stiffned plates. Comput. Structures 8, 51-60 (1978). 61. A. Peano, Conforming approximations for Kirchhoff plates and shells. Int. J. Num. Meth. Engng 14, 1273-1291 (1979). 62. R. Courant, Variational methods for the solution of problems of equilibrium and vibrations. Bull. Am. Math. Sot. 49, l-23 (1943). 63. R. J. Melosh, Basis for derivation of matrices for the direct stiffness method. Am. Inst. Aeronaut. Astronaut. J. 1 (7), 1631-1637 (1963). 64. B. Fraeijs de Veubeke (Ed.), Upper and lower bounds in matrix structural analysis. Matrix Methoak of Structural Analysis, pp. 165-201. Pergamon Press, Oxford (1964). 65. T. H. H. Pian, Derivation of element stiffness matrices by assumed stress distributions. Am. Inst. Aeronaut. Astronaut. J. 2 (7), 133-1336 (1964). 66. L. R. Herrmann, Finite element bending analysis for plates. J. Engng Mech. Div., ASCE 93, (EM5), 13-26 (1967). 67. S. W. Papenfuss, Lateral plate deflection by stiffness

493

matrix methods with application to a marquee. Thesis presented in partial fidfilhnent of the requirements for the degree of Master of Science, Department of Civil Engineering, University of Washington, Seattle, Washington (1959). 68. 0. C. Zienkiewicx and Y. K. Cheung, The tinite element method for analysis of isotropic and orthotropic slabs. Proc. Inst. Civil Engrs, London, Vol. 28, pp. 471-488 (1964). 69. D. J. Dawe, A finite element approach to plate vibration problems. J. Mech. Engng Sci. 7 (I), 28-32 (1965). 70. R. T. Sevem and P. R. Taylor, The finite element method for flexure of slabs when stress distributions are assumed. Proc. Inst. Civil Engng, Lot&n 34, 153-170 (1966). 71. R. D. Henshell, D. Walters and G. B. Warburton, A new family of curvilinear plate bending elements for vibration and stability. J. Sound Vib. XI (3) 381-397 (1972). 72. R. D. Henshell, On hybrid finite elements. The Mathematics of Finite Elements and Applications (Edited by J. R. Whiteman), pp. 299-311. Academic Press, London (1973). 73. G. A. Butlin and F. A. Leckie, A study of finite elements applied to plate flexure. Presented at The Symp. on Nwn. Meth. for Vib. Problems, University of Southampton (July, 1966). 74. H. Hansteen. Finite element disnlacement analvsis of plate bending based on r&angular ele&nt~ Presented at The Symp. on the Use of Electron. Digital Comput. in Structural Engng, University of Newcastle upon Tyne (1966). 75. V. Mason Rectangular elements for analysis of plate vibrations. J. Sound Vib. 7 (3), 437-448 (1968). 76 N. Poonlewell and D. McDonald. Conforminn rectangular-and triangular plate-bending elements. 3. Sound Vib. 19, 333-347 (1971). 77. S. Gopalacharyulu, A higher order conforming, rectangular plate bending element. Int. J. Num. Methodr Rngng 6 (2), 305-309 (1973). 78. D. S. Watkins, A conforming rectangular plate element. The Mathematics of Finite Elements and Applications (Edited by J. R. Whiteman), pp. 78-83. Academic Press, London (1975). 79. D. S. Watkins, A comment on Gopalacharyulu’s 24 node element. Int. J. Num. Meth. Engng 10 (2), 471-474 (1976). 80. D. J. Dawe, On assumed displacements for the rectangular plate bending element. The Aeronaut. J. R. Aeronaut. Sot. 71, 722-724 (1967). 81. B. E. Greene, R. E. Jones, McLay R. W. and D. R. Strome, Generalized variational principles in the finite element method. AIAAIASME 9th Structures, Structural Dynamics and Materials Co&, Palm Springs, California (April 1968). 82. T. H. H. Pian and P. Tong, Rationalization in deriving approach. Proc. 2nd Conf. on Matrix Meth. in Structural Mech., pp. 441469, Ohio (Oct. 1968). 83. T. H. H. Pian, Hybrid Models. Numerical and Computer Methods in Structurat Mechanics (Edited S. J. Fenves et al.), pp. 5978. Academic Press, New York (1973). 84. T. H. H. Pian and S. T. Mau, Some recent studies in assumed stress hybrid models. Advances in Computational Methods in Structural Mechanics and Design (Edited by J. T. C&n et ai.), pp. 87-106. UAH Press, Alabama (1972). 85. I. Holand, Membrane and flat plate elements. Proc. World Cong. on Finite Element Meth. in Structural Mech., Boumemouth, Dorset, England, Vol. II, pp. 13.1-13.18 (1975). 86. I. M. Smith, A 6nite element method analysis for

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“moderately thick” rectangular plates in bending. Znf. J. Me&. Sci. 10,563-570 (1968). 87. I. M. Smith and Duncan W. The effectiveness of excessive nodal continuities in the finite element analysis of thin rectangular and skew plates in bending. Znt. J. Num. Meth. Engng 2 (2), 253-257 (1970). 88. A. W. Wegmuller and C. N. Kostem, Finite element analysis of plates and eccentrically stiffened plates. Fritz Engng Lab. Rep. 378 A.3, Lehigh University, Bethlehem, Pennsylvania (1973). 89. A. W. Wegmuller, Finite element analysis of elastic-plastic plates and eccentrically stiffened plates. Thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy, Department of Civil Engineering, Lehigh University, Bethlehem, Pennsylvania (1971). 90. A. W. Wegmuller, A refined plate bending finite element. Int. J. Solids Structures 10 (ll), 1173-1178 (1974). 91. A. Poceski, A mixed fmite element method for bending of plates. Znt. J. Num. Meth. Engng 9 (l), 3-15 (1975). 92. A. Razzaque, Finite element analysis of plates and shells. Thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy, Department of Civil Engineering, University College, Swansea (3972). 93. R. A. F. Martins and D. R. J. Owen, Thin plate semi-loof element for structural analysis including stability and natural vibrations. Znt. J. Num. Meth. Engng 12 (ll), 1667-1676 (1978). 94. E. D. L. Pugh, The static and dynamic analysis of Mindlin plates by isoparametric elements. Thesis presented in partial fuffillment of the requi~ments for the degree of Master of Science, Department of Civil Engineering, Department of Civil Engineering, University College, Swansea (Dec. 1976). 9.5. T. J. R. Hughes and M. Cohen, The “Heterosis” Iinite element for plate bending. Proc. ASCE Conf on Electron. Computation, pp. l-10 (Aug. 1979). 96. M. M. Hrabok and T. M. Hrudey, Analysis of stiffened plates by the hybrid stress finite element method. Structural Engng Rep. 100, Department of Civil Engineering, University of Alberta, Edmonton, Alberta (Ott, 1981). 97. A. Chatterjee and A. V. Sethu, A mixed finite element formulation for plate problems. Znt. J. Num. Me&. Engng 4 (l), 67-84 (1972). 98. R. J. Melosh. A flat triangular shell element stiffness matrix. Prac.~ 1st Conf. CI; Matrix Meth. Structural Mech., Ohjo, pp. 503-514 (Oct. 1965). 99. S. Utku, Stiffness matrices for thin rectangular elements of non-zero Gaussian curvature. Am. Inst. Aeronaut. Astronaut. J. 5 (9), 1659-1667 (1967). 100. S. Utku, Gn derivation of stiffness matrices with C” rotation fields for plates and shells. Proc. 3rd Conf. on Matrix Meth. in Structural Mech., Ohio, pp. 255-274 (Oct. 1971). 101. H. C. Martin, Stiffness matrix for a triangular sandwich element in bending. TR 32-l 158, Jet Propulsion Lab., Pasadena, California (Feb. 1968). 102. G. Dhatt, Numerical analysis of thin shells by curved triangular elements based on discrete Kirchhoff hypothesis. Proc. Symp. on Application of Finite Element Meth. Civil Engng, School of Engineering, Vanderbilt University, Nashville, Tennessee, pp. 25.5-278 (1969). 103. G. Dhatt. An efficient triangular shell element. Am. Inst. Aer&aut. Astronaut. J. $ (1 l), 2100-2102 (1970). 104. I. Fried and S. K. Yang, Triangular 9 degree of freedom, C? plate bending elements of quadratic accuracy, Quart. Appt. Math. 31 (3), 303-312 (1973). 105. E. Hinton, A. Razzaque, 0. C. Zienkiewicz and J. D. Davies, A simple finite element solution for plates of homogeneous, sandwich and cellular construction. Proc. Inst. Civil Engrs, London, Vol. 59, Part 2, pp. 43-46 (March 1975).

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495

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