Finite Element by Tarun Kant

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Compua~ b Smcr wer Vol .15.No. 2. ~. 177483, 982

F%wdi nGrer l Bri l ai n.

A REFINED HIGHER ORDER Co

PLATE BENDING ELEMENT

TARUNKANT+

Department of Civil Engi neering, Indian Institu te of Technology, Bombay 400 076. India

and

D. R. J. OWENS nd 0. C. ZIENKIEWICZQ

Department of Civil Engi neering, University of Wales, University College of Swansea. Swansea SA2 8PP. Wales

(Received 2 December 1980; received for publication 20 lanuory 1981)

Abstract-A general finite element form ulation for plate bending problem based on a higher-order displacement

model and a threedimensional state of stress and strain is attempted. The theory in corporates linear and quadratic

variations of transverse normal st rain and transverse s hearing strains and stresses respectively throu gh the

thickness of the plate. The 9-noded quadrilateral from the family of two dimensional Co conti nuous isoparametric

elements is then introduced and its performance is evaluated for a wide range of plates under uniformly distr ibuted

load and with different support cond itio ns and ranging from very thick t o extremely thin situations. The effect of

full . reduced and selective integration schemes on the final numerical result is examined. The behaviour of this

element with the present for mulation is seen to be excellent under all the three integration schemes.

INTRODUCTIOh

Isoparametric plate elements [ I-51 based on Mindlins

theory(61. which obviate most of the problems that beset

elements based on the classical Kirchhoffs plate

theory[7]. have become popular and well established.

However the theory itself has certain limitations, viz.,

the effects of transverse normal strain and transverse

normal stress are not accounted for. the transverse

shearing strains are assumed constant through the

thickness and a fictitious shear correction coefficient is

introduced to account for warping of the cross-section.

Thus the claim that the element is applicable to both thin

and thick plates is not tenable. At the most only

moderately thick plates could be analysed. In this paper

a Co continuity element is developed based on a higher-

order displacement model than hitherto used and a three

dimensional state of stress and strainI8.91. Specifically,

the transverse normal strain and the transverse shearing

strains are now assumed to vary linearly and quadratic-

ally respectively throu gh the thickness

of the

plate. Thus

the warping of the cross-section is automatically in-

corporated. This element retains the simplicity of

Mindlins fermulation and. at the same time, has no

intrinsic limitations. It is expected that this development

will offer refined and realistic solutions in the case of

really thick situations in general, and composite/layered

systems, stress-distribution near concentrated loads, etc.

in particular. The element which has six degrees of

freedom per node can employ the available family of the

Serendipity and the Lagrange elements[S, IO]. In the

present study a 9-noded Lagrange .quadrilateral is used

which has performed very well on all of the test problems.

Numerical results corresponding to exact, reduced and

*Assistant Professor. presently a Visitor in the Deoartment of

Civil Engineering. University -of Wales, University College

Swansea. Swansea SA2.8PP. Wales. under the Jawaharlal Nehru

Memorial Trust (U.K.1 Scholarship 1979 Award.

*Reader.

#Professor and Head of Department.

OUfx, y, z), 6,(x, y), w*(x, ). etc. will henceforth be written

simp ly as 1. 6,. a+*. etc.

selective integrations for a wide spectrum of plates-very

thick to extremelythin-are presented.

ELEMENT FORMULATION

The general theoryll, 91 is based on the displacement

model (Fig. 1)

Lltx. y, I)= z&(x, y)+ z3B$tx,y)

u =

L

vtx. y. zl = 20,(X, y) + z??:tx, y)

wtx. , 21 = wtx, y) + z2w*tx, y)

m

where U. 1 and W define the displacement components

in the coordinate directions x, y and z respectively. Both

the x and y coordinates lie in the reference plane of the

plate which is assumed unstrained. The terms 8, and 6,

are the usual average rotations of the normals to the

reference plane along the x and y directions respectively,

while w defines the transverse displacement at the

reference surface. The terms @, 8: and w* are the

corresponding higher-order terms in the Taylors series

expansion used in the present theory and are also defined

at the reference plane. Thus the generalized displace-

t

z

W

8,

V

)_

c

Y

J

Fig. I. Positive set of displacement components.

177


2/7

178

T. KANT et ai.

m nt

f the reference plane S is expressed in terms of

six independent variables,

s = [w, RX,0,. w*, et, 0:).

(2)

The six three dimensional strain components are then

related to the generaiised displ acement compo nents 6 by

the relation,

E

1[

-K +zK*

. I

Q, = zK, + zK;

Y

yxy = zK,, t

z'K:,

Yxz d, z24 ,

yy* d, Z'bC

E,=zzw*

(31

where the new curvature terms K,, KY, etc. called the

generalised strain components vector Q in the two

dimensi onai fini te element context , are related to the

generalised dis placement components 6 by the following

matrix relation.

o~oooo-

0

0 0 0

Od&$O

0 0

~10000

~01000

0000~0

A

00000;;;

oooo~-&

000;30

000;03

oooioo

= L6 (4)

The total p otenti al energy 1~ for the present theory[9)

is given by

rrudA-

j-/z++p.-l(w+;w*)dA

(5)

where pz+ and pL- are the transverse dist ribu ted loads on

the posit ive and negative extreme z planes respectively

and h is the total th ickness of t he plate. The generalised

stress component vector (I which is the integral of the

physical str ess components through the thickness of the

plate (see Ref. (91 and Figs. 2-4) is given by

v = IMP. M,. M,,, Qx, ,. L M:, d:,,:.Q:,AT.

6)

The generalized stress vector u and the generalised

strain vector E are partitioned as follows:

and

where

oh = IM,, M,, MrvIT. ,,z = [M:, M:,

M:,],

@1,3= Mz,a,, =[Qx,

Qy17.u,z=[Q:.Qtl'.

+,t Kc. Ky, Kxy]. e,,z = [KC. KF. K:,],

43

2w*. f,I = [&. &I. e.2 = [bT, C#Jt].

(9)

For a linear elastic material the constitutive relation c an

be writ ten as

ghl = ,rb

+ hZChZ + Qb)h3 zw*

1 IOal

a h 2 E ~, f D d h 2 f Zw

c ob,

gh h) = 4, ,, i - D&h2 + &, ?W*

f IOCJ

~~I=QslE,i~D,?E,:

ClOdl

ff,: = &%,I + 4&? 1 Oel

where the elasticit y matrices 4,. Q,,*. &,, pVz. etc. for a

homogeneous and isotropi c plate with Youngs modulus

E. Poissons ratio v and thickness h are expressed in the

following manner:

&z= Eh5

8O(l + v)(l -2~)

?h)h4 4,8(1

Eh

+ V)(l-ZV)

I)h.(= Eh5

8Ofl + u)(l -2~)

Eh

= 12(1 +v)(l-zV)

(I - VI

(IIf)

( I Ig)

(llhl

Eh IO

DS, =-

i 1

6Otl+v) 0

I

Illil

IL -1 0

0

0

I -2v

,

IL

0

I

1

?v

i1

L 0

I o

0

--J0

I-2V

2

(I la1

illb)

I I IC)

(IId)

IlIe,


3/7

A refined higher-order Co plate bending element

179

?_

2

I/

F

Y

2

\Reference

surface

Fig. 2. Positive set of stress components.

/ / ,i

(P,*i,) /

/ / /

Fig. 3. Positi ve set of stress resultants-forces

dy

CAS

Vol

IS.

No.2-F

Fig. 4. Positive set of stress resultants-couples


4/7

180

T KANT t

al.

It is seen that the strain energy expression of eqn (5)

contains only the first derivatives of the components of

the generalized displacement vector 6 and thus only Co

continuity is required for the shape functions to be used

in the element formulation. If the same shape function is

used to define all the components of the generalized

displacement vector S, then

6 = f

NiSi

i-l

(12)

where N, is the shape function associated with node i, &

is the value of 6 corresponding to node

i

and m is the

number of nodes in the element.

With the generalized displacement vector 6 known at

all points within the element. the generalized strain vec-

tor Q at any point is determined with the aid of eqns (4)

and (12). as follows:

where

Bi = LN,

(14)

and is given explicitly for the present case as follows:

,

0~0000

oo;ooo

o;;ooo

~l0000

~01000

0000~0

00000~

OOOOLL

dy

b

oooi30

000;03

000200

V,

(IS)

Having obtained the D and 8 matrices as given by

eqns (1 I) and (IS) respectively the element stiffness

matrix 0. can be conveniently expressed in the follow-

ing form [ IO]:

K=

BTQB dA =

BTDBIJI d5 d?

(16)

For the present element which has six degrees of

freedom per node as opposed to Mindlins element[I, IO]

which has only three degrees of freedom per node, it is

considered appropriate to economise on the cost of

computation involved in numerical quadrature for

evaluating the integral in eqn (16). It is generally agreed

that the foregoing computation accounts for a very sub-

stantial part of the total computation cost of the com-

plete analysis[lll. To achieve this end, the element

stiffness matrix is partitioned in both the row and column

directions into blocks. Each block corresponds to the

degrees of freedom at a node[12]. The expression of the

stiffness coefficients for a typical block ij could be writ-

ten as follows:

K = i,

2,

WPw,IIBiTm,

(17)

where W and W are weighting coefficients, n is the

number of numerical quadrature points in each direction

and & and & are the strain-displacement matrices based

on the ith and jth shape functions, respectively. Due to

symmetry of the stiffness matrix K, only the blocks K

lying on one side of the main diagonal are formed (Fig.

5). Further, the suggestion of Gupta[l3] is incorporated

in the program which stipulates explicit multiplication of

the Bi, D and tj matrices instead of carrying out the full

matrix multiphcation of the triple product. This is

expected to reduce the computing time for element

stiffness formulation by a factor of nine. For the con-

venience of any prospective user, these 6 x 6 submatrices

corresponding to flexure and transverse shear effects are

presented in Appendix I.

The consistent load vector P due to distributed load p

can be written in general as:

P=

I

+I t,

NTpdA =

A

IJ

Nrp//l dl dq (18)

-I -I

which takes the following form suitable for numerical

integration when transformed in the context of the

energy expression (5) for the theory presented in this

paper

I

1

:

p=pi, J, w~w~lzWi p14

(A+ f P*-).

(19)

S _

0

I 0

It is noted that eqn (19) is different from the usual ones

because the loads are not assumed to act on the

reference surface in the present development.

I

,

I

$*

I Kz3 ;_-_--_________J K2m

1 -

I-

I

I

I

Symmetrical ,

I

\

I

\ I

\

\

\

cl_

mm

m: number of nodes

1

Fig. 5. Nodal partitions of the element stiffness matrix.


5/7

A refinedhigher-orderCo plate bendingelement

181

A computer program was developed based on the

foregoing approach to generate the element stiffness

matrices, the consistent load vector and a few related

processes for the stress compu~tion. This program was

integrated with the modified version of the one described

in detail by Hinton and Owen[l4].

NtlME3UCAl E-

A square plate of side a = 1 and subjected to a trans-

verse distributed load p2 = 1 is analysed with three types

of ~undary conditions. A Poissons ratio (v) of 0.3 is

assumed. The numerical experimentation is carried out

with four of the nine noded Lagrangian quadrilateral

elements in a quarter plate. All the computations were

performed on a CDC 7600 computer in single precision

with 14 significant digits of real rounded arithmetic ac-

curacy. Full. reduced and sefective integration schemes

based on gauss-Legendre product rules, viz. 3 x 3 and

3 x 3f.E). 2 x 2 and ? x2(R) and 3 x 3 and 2 x 2(S), have

been employed for flexure and shear contributions res-

pectively to the element stiffness matrix computation.

The results for the plate analysed with the three boun-

dary conditions are presented in Tables 1-3 for various

values of the aspect ratio u/h. Typical execution time is

0.52 CP seconds per analysis.

CONCLUSIONS

The performance of the 9-noded Lagrangian iso-

parametric element has been studied in conjunction with

the refined higher-order plate theory presented in this

paper-and also elsewhere[8,9]. On the basis of the

results of Tables l-3 it can be positively concluded that

the elements buoyance is excelient in all the situa-

tions studiid in this paper with no apparent preferential

choice for full (exact), reduced and selective integration

schemes for computation of the element stiffness

matrix[l-5,15,16]. This appears to be very significant

and perhaps is a pointer towards the use of the present

relined higher-order theory in preference to Mindlin

theory hi~erto used in most of the analyses. Qualita-

tively this could be due to the better represen~tions of

the cross-sectional deformations and the stress-strain

law. However, its behaviour in context of the few boun-

dary constraints which lead to the development of

mechanisms (or near mechanisms) commonly interpreted

as zero energy modesf5f needs to be seen.

REFERENCES

E. Hinton. A. Razzaque.0. C. Zienkiewicz nd J. D. Davies.

Simple finit e element solut ion for plates of homogeneous,

sandwich and cellular cons tructi on. Proc. of the Insfitutian

of Ciuil Engineers, Pan Ii. 59.4365

(1975).

T, J. R. Hughes. R. L. Taylor and W. Kanoknukulchai.A

simple and efficient fini te element for plate bending . inr. j.

Num. Me&. Engng. 11, 15294543 1977).

E. D. L. Pugh E. Hinton and 0. C. Zicnkiewicz. A study of

quadrilateral plate bending elements with reduced in-

tegrat ion . Int. 1. Num. MeGr. Engn g. 12. 10594079 (19781.

Table I. Simply supported

quare plate under unifo rm loading with w = @,= W* = f?: = t&l along its four edges

a/h 1ntes.

centI-

Central+

tWll.r+ flid-edge+

TYW

01s01acemant

Bendin Ploment

s

T\,lst ing l%rnent Shear Force

P&?/O

Pa

Pa

Pa

E

0.00653

0.0521 - 0.0223 0.311

2 R o.GG054

0.0509 - 0.0216 0.285

S

O.OOB5~

0.0522

- 0.0219 0.285

E 0.00612 0.0499 - 0.0276 0.315

3 Fi 0.00612 0.0488 - 0.0262 0.285

s 0.00611 0.0500 - 0.0272 0.265

E 0.00522 0.0491 - 0.0295 C.322

4 R 0.00523 O.D4RD - 0.0273 0.285

s 0.00523 0.0492 - 0.0292 0.285

E O.OG480 0.0467 - 0.0304 0.330

5 R 0.00482 0.0477 - 0.0267 0.285

S 0.00481 0.0488 - 0.0302 0.285

E 0.00423 0.0480 - 0.0315 0.381

IO R 0.00426 0.0472 - 0.0299 0.285

s 0.00426 0.0484 - 0.0314 0.2&S

10.00427 Exact;

E 0.00398 0.0470 - 0.0317 0.573

5c R G .00408 0.0471 - 0.0302 G.ZBS

5 0.00408 0.0482 - 0.0319 0.285

E 0.00397 0.0469 - 0.0317 0.597

1OC R 0.00407 0.0471 - 0.0302 0.205

5 0.00407 0.0482 - 0.0319 0.264

E 0.05396 0.0468 - 0.0317 0.606

1OOG R 0.00407 0.0473 - 0.0302 C.285

S c .0040? 0.0482 - 0.0319 0.234

E 0.0039E

0.0468 - 0.0317 O.SOE

10000 R C.OO4G7 0.0471 - 0.0302 0.285

s 0.00407 0.0482 - 0.0319 0.2P4

I

I*+

0.004OE

0.0479 - 0.0325

0.338

1

Classic+? thin piate solution [Fief. 71


6/7

T. KANT et ai

Table 2. Simply support ed sq uare plate under uniform ioading w ith w = W* = &s*t along its four edges

a,h In-%.

Central

cm r41+ corner+

Rid-edBe+

Type

01sp1acsmnt

Bendi ng mmm Tw st i n FI omant

4

Shosr FOTCD

' pa' / 0

*Da* *XI

*Pa

E 0.00949 0.0602 - 0.00206

0.324

2 R 0.00944 0.0596

-

Q.00302 0.30t

S 0.00943 0.0605 - 0.00095 0.301

E

0.00605 0.0573 - 0.00509 0.336

3 R 0. 00893 0. 0566 - o.oo545 0.310

S 0.00693 0.0577 - 0.00337 0.309

E 0.00592 5.0552 - 0.00760 0.345

4 R 0.00593 0.0548 - 0.00751 0.314

S 0.00592 0.0556 - 0.00536 0.312

E 0.00536 0.0536 - 0.01051 0.353

5 R 0.00541

0. 0534 - 0. 00026 0. 316

s 0. 00540 0. 0546 - 0. 00716 0. 313

E 0. 00446 0. 0503

-

0. 02Oc 0. 397

10 A 0. 00456 5. 0503

-

0. 0152 0. 322

S 0. 30456 Cl . 0521

- 3.0135

0. 316

E o. I 30400 3.0472

-

0. 0306 0.561

50 R 0. 00419 0. 0483 - 0.0218 0. 341

S 0. 00420 0. 0511

-

a. 0216 0.339

E 0.00307 0.0469

-

0. 0314 0. 563

100 A 0.00416 I l . 0482 - 0. 0221 0. 343

s 0. 00416 0. 0511

- 0.0221

0. 341

E 0. 00396 0.0466 - 0.0317 0. 591

1000 R 0. 00417 0. 0461

- 0.0223

0. 344

S 0.004' 17 0.0511

-

0. 0223 0. 342

E

0.00396

0. 0466

-

0. 0317 3. 591

10000 R 0. 00417 3. 046

- 3.0223

3. 344

5 0. 00417 G. 0511 - 0. 0223 0. 342

+

val ues U. neareat Gauss poi nt.

Table 3. Clamped square plate und er unif orm loading with w = l+ = 8, = w* = 8: = 6: = o(c) alon g its four edges

a/h

Intag.

Centr al C&WC

ilid-Edge+

nid-ea*~+

: YW

Ofsplacment Bendi ng mnent Bendin i %moot

f

Shssr Fores

- pa' / D - 5. 2

*Pa a

E 0. 00607 0. 0335

-

0.0262 0.332 . -

2 R 0. 00611 0. 0320

- 0.0100

0. 297

s J . 00609 0.0338

-

0. 0267 0.297

E 0. 00349 0.3263 - 0. 0300 0. 352

3 R 0. 00355 0. 0270 - 0. 0242 0. 306

S 0. 00352 0.0267 - 0. 0319 0. 306

E 0. 50253 0. 3261

-

0. 0320 0.374

4 R C. 00261 0. 0252

-

0. 0264 0.315

5 0. 00256 0. 0267

-

0. 0344 0.315

E G. 00206 0. 0250

-

0. 0337 0.396

5 R 0. 00216 0. 0243 - 0. 0277 0.323

s 0. 00211 0. 0256

-

0.0350 0. 323

6 0. 00136 D*O226 - o. bw 0. 516

10 R 0. 00152 0. 0226

- 0.0299

0. 344

5 0. 00146 0. 0236

-

0. 0387 0.344

E 0.001w

0. 0207 - 0. 0194 0. 906

so R 0. 00129 0. 0222

-

0. 0308 0.356

S 0. 00123 0. 0227

-

0. 0309 0.356

E 0. 00097 0. 0205 _ 0. 0177 0.956

100 R 0. 00126 0. 0222 - 0. 0308 0. 357

S 0. 00123 0. 0227 - 0. 0390 0. 356

E 0. 00096 0. 0204

1000 R 0. 00126 0. 0222

S 0. 00222 0. 0227

- 0. 0170

0. 975

- 0.0308

0. 357

- 0.0390

0. 356

E 0. 00096 0. 0204 - 0. 0170 0. 975

10000 R 0. 00123 0. 0222 - C.0308 0. 357

S 0. 00122 0. 0227 - 0. 0399 0. 356

: 1

i t

0. 00126

0. 0231

- 0.0513

+

t+

Cl assi cal thi n pl ate sol uti on [Ref. 71.


7/7

A refined higher-order Co plate bending element

183

4. T. J. R. Hughes, M. Cohen and M. Haroun. Reduced and

selective integration techniques in the finit e element analysis

of plates. Nucl. Eng ng Design 46.203-222 (1978).

5. T. J. R. Hughes and hf. Cohen, The Heterosis fini te ele-

ment for plate bending. Compuf. Structures 9. 44.5-450

(1978).

R. D. Mindli n, Influence of rotary inertia and shear on

flexural motion s of isoto nic elastic plates. 1.

ADD/. Mech. 18.

31-38 (1951). ..

S. P. Timoshenko and S. Woinowsky-Krieger. Theory of

Plates and Shells. McGraw-Hil l. New York (1959).

E. Reissner. On transverse bending of plates, inclu ding the

effects of transverse shear deformation. Int. .I. Solids Struc-

tures 11,569-573 (1975).

T. Kant, The segmentation method in the analysis of elastic

plates with two oppo site edges simply support ed using a

refined higher order theory.

Rep. C/R/366/80.

Department of

Civil Engin eering, University of Wales. Swansea SA2 8PP,

1980 (to be published).

IO

II.

12.

13.

14.

15.

16.

0. C. Zienkiewicz. The Finite Elemenf Method. McGraw-Hill,

U.K. (19771.

Y. K. Cheung

Finite E~emenf

and M. F. Yeo, A Practical Introduction to

Analysis. DD. 94 . Pittnan, London (1979).

A. K. Noor and S. J:Har&y. Evaluation of element stiff ness

matric es on CDC STAR-100 comp uter.

Compuf. Structures

9. 151-161 (1978).

A. K. Gupta and B. Mohraz. A method of comput ing

numerically integrated stiffness matrices. Int. 1. Num. Mefh.

Engng. 83-89 (1972).

E. Hinton and D. R. J. Owen, Finite Element Programming.

Academic Press. Lon don (1977).

0. C. Zienki ewicz. R. L. Taylor and J. M. Too, Reduced

integration techniques in general analysis o f plates and

shells. Inf. 1. Num. Meth. Engng. 3, 275-290 (1971).

0. C. Zienkiewicz and E. Hinton. Reduced integration, f unc-

tion smoothing and non-confor mity in finit e element analysis

(with s pecial reference to thick plates). L Franklin Inst. 302,

443-461 (1976).

tat Flexure contribufion.

41,-2.i

N. 9

0

x,, I-2,

-

N,N

0,11-2.1

.?

2

- N;.dN,

0

z

p,

5

Dell-2.) N, N,

5D,W2r)N,3 +&(I-Zv)N,N,

0

+DD:ll-2v)N. 9

D:(I-2v) N,N,

0

D,(I-2) N, 9

0

+D,(I-2u) N, N,

4

Dfil-2~1 N. N,

fb) Transverse shear contr ibuti on.

tK, = _Eh/(l + v)(l - 2s). D, = K,h%2. of = 3D,h2/20, D:* = 5mh2/28

Finite Element by Tarun Kant

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