Four Node Plate Bending Element

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0045-7949(94)00351-3Copyrighl i 1995 Elsevier Science Ltd

Prmted m Great Brimin. All rights reserved0045-7949195 19.50 + 0.00

A NEW FOUR NODE QUADRILATERAL PLATEBENDING ELEMENT

0. L. Roufaeil

7 Fischer Court, Toowoomba, Queensland 4350, Australia

(Received 2 1 August 1993)

Abstract A new four-node quadrilateral plate bending element is presented for the analysis of thin andmoderately thick plates of general plan form. The element has three degrees of freedom at the corners.Numerical experiments are given and they indicate that the element does not lock for thin plates andgives excellent results for both thin and moderately thick plates.

1 INTRODUCTION

The development of the early plate bending elementswas based on the classical plate theory (CPT). Be-cause it was difficult to achieve the continuity of thedeflections and their slopes (i.e. C continuity), the

focus was shifted to Mindlin [l] and Reissner platetheories [2]. The advantages of using these theoriesare: they require displacements to be continuous (i.e.Co continuity) and they are more accurate in hand-ling composite materials, where the transverse sheardeformation is very significant. A review of many ofthese elements is available in [3-61.

The elements, which are based on Mindlin andReissners theories, work well for thick plates. How-ever, in some cases as the plate thickness decreases theshear term dominates the stiffness matrix and causeswhat is known as the shear-locking phenomenon.

To overcome this problem it becomes necessary touse the reduced integration or selective integrationtechniques. On the other hand if the adopted order ofnumerical integration is too low, the stiffness matrixwill be such that certain deformation patterns (otherthan rigid body modes) will involve zero strain energyand lead to a possible mechanism. One of the earlyapproaches for analysing the moderately thick shellswas introduced using some isoparametric shell el-ements [7]. However, it was only the use of reduced[8], and selective [9] integration that allowed thin aswell as thick plate behaviour to be modelled accu-rately with the eight-noded element. Other ap-proaches for Mindlin plate elements based on penaltymethods and mixed formulations have been proposedto solve the shear-locking problem. Reviews andexplanations of these techniques are available in[3, 51.

By comparison with CPT, relatively few analyticalsolutions based on Mindlin plate theory have been

reported. However, some numerical solution schemesbased on finite difference [lo], finite strip [l 1] andRayleigh-Ritz solution [ 121 have been presented.

The objective of this paper is to present an efficientquadrilateral four-node element with 12 degrees offreedom for plate bending. This element is based on

Mindlin plate theory (MPT). The shape functions ofthe displacement and rotations are not completelyindependent. Some examples and tests are given toshow the behaviour of this element. Numerical resultsare presented for problems involving rectangularplates of different aspect ratios and support con-ditions, skew and circular plates. The present elementperforms quite well for the class of problems studiedand does not show any sign of the shear-lockingproblem.

2 MINDLIN PLATE ELEMENT FORMULATION

2.1. FundamentaI relations

In Mindlin plate theory a straight line originallynormal to the plate median surface will be straightbut not normal after deformation. In reality, such aline is generally curved but the assumed straight,non-normal line can clearly approximate this in anaverage sense at all points. A shear factor is intro-duced to take account of the fact that the shear straindistribution through the plate thickness is not uni-form. The inclusion of transverse shear effects in theplate theory means that the cross-sectional rotationsY, and YJ are no longer expressed solely in terms ofthe deflection W. Thus, there exist three basic refer-ence quantities, namely W, Y, and Y, , at the mediansurface rather than simply w alone in the CPT.Figure I shows the positive directions of W, Yy, andYy,..

871


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872 0. L. Roufaeil

2.3. Stress-strain relations

For an isotropic material, we have the following

/

Ytwo stress-strain relations.(a) The momentcurvature relationships are

Fig. 1. Displacement sign conventions used in Mindlin platetheory.

Following Mindlins assumption [l], the displace-ments of any point in the plate U, V and W areexpressed, as

U(x, y, 2) = --zY,(x, Y)

~(-?y,z)= -ZY,(X,Y)

wx, Y, 2) = 4x, Y). (1)

where

and

(3)

where w, Yy,, Yy are translation and the normalE is Youngs modulus, v is Poissons ratio and h is the

rotations of the midplane, respectively. The rotationsplate thickness.

Y, and ul,. can be expressed in the form(b) The shear force-strain relationships are given

as

and Y,,=F-yJz,ay (4)

where awlax and aw/ay are the slopes in the x andy directions of the deformed median surface and yIz

where

and Y,,.~ re the shear strains.@:

2.2. Strain-displacement relation

For MPT, the strains on the middle surface areand

given in [I] as follows:[D,]=K2Eh

~Tz_~[(1 -v) 0 1

where K* is the shear constant.

3 FINITE ELEMENT FORMULATION

3.1. Element d i splacement fi el d

An element El6 with 16 degrees of freedom, asshown in Fig. 2(a), was proposed by the author in[13]. The displacement functions of this element aregiven below

aIW

(i [x y xy x* y* x*y xy* x3 y3 x3y xy3 0 000 a2

Yy, = 0 0 0 0 2x 0 2xy y* 3x2 0 3x*y y3 1 ~00

Yy, 000 0 0 2y x2 2xy 0 3y* x3 3xy* 0 01.x

ll a3

;

aI6

(5)


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(a)

A new four-node quadrilateral plate bending element 873

t T

(b)Q3

i.i

c _..7I?2

r 239..

LI?121_a4 Yrz4 a1. .-(cl

Fig. 2. The plate element. (a) Degrees of freedom of element E16. (b) The proposed element includingthe shear degrees of freedom. (c) A Timoshenko beam element.

The element used in this work has the same shape The vector {u} which may be expressed in terms offunctions as E16, but Y,; will be introduced at the the displacements ismidsides as shown in Fig. 2(b). From equation (2) wecan see that (0) = Pl(~ *It (9)

where [P] is a matrix of order 16 x 16 for this elementY,5z=(~-~,)cosx+~~-P,)sinr. (6) and

where

and

x - xcos a = __

11,

sin t( = +J,

The above element has 16 degrees of freedom; threedegrees of freedom at each corner and one degree offreedom at the middle point of each side of theelement, as shown in Fig. 2(b). Each side will beconsidered as a Timoshenko beam element, as shownin Fig. 2(c). Therefore, each midside degree of free-dom, e.g. Ysr,, may be expressed in view of eqn (6) asfollows:

(see Fig. 2~). WI - w2 4h1 + ti,2Now, the fundamental quantities, W, Yy,, (v,. and YS._ YCI=

,,I + *r2 .- - ____ cos a, - A sm a,I 2 2

may be written as follows:12

(10)

W

li 1x Y XY x2 y2 xy xy2

YY 00 0 2x 0 2xy y ?=, 0 0 0 0 0 2y x2 2xy

YC 0 cosa sina ycosafxsi na 0 0 0 0

x3 y x y xy 0

3x2 0 3x:y y3a2

1 Y 0 0X a23y2 x3 3xy2 0 0 I

i/:1l.

X(7)

0 0 0 0 -cos a -y cosa -sin a

Equation (7) may be cast in the following form

= P(.v, v)l{a}.

Using the above relation, the vector {S *) of eqn (9)may be expressed in terms of the corner degrees of

8) freedom only, {6}, as follows:

is*> 16xI=~H1,6x,2~6}12xI, (11)


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874 0. L. Roufaeil

where

I

1-_H*

where w], {6}, {f } re the element stiffness matrix,displacement vector and the element forcing function,respectively.

In eqn (14) the element stiffness matrices werecalculated using 4 x 4 and 2 x 2 Gauss quadrature

[HI=

[I] is the identity matrix of order 12, p*] is a 4 x 12 schemes for the bending and the shear strain energies,matrix, and respectively.

During assembling the element stiffness matrices

(61 12X = {w~+YIrti?.I~ W2, > W47IC1,4> ,.41T. into the global stiffness matrix, the effect of thesuppressed degrees of freedom, e.g. y,:,, must be

From eqn (10) the matrix [H*] is expressed as carefully dealt with. Although these degrees of free-follows: dom have been suppressed, their effect still exists. As

1- - cosa,

1

I-isinE, -- -;cos a, -fsinu,

I2 I I2

1

p-I*]= O O0

a

-jcoscc, -fsina,

0 0 0 0 0 0

I--

1-icosa, -tsincc, 0 0 0

41

0 0 0 0 0 0

1-I

-;cosa, -fsina, 0 0 023

I 1

G-;cos aj -fsina, --

1-4cos a, -fsina,

341

0 0 0 -1 -;cosGL4 -fsincz,41

(12)

~1,a2,..., a4 are shown in Fig. 2(b) and l,> is the shown in Fig. 3 these degrees of freedom as vectorslength of the side between nodes 1 and 2 and similarly must be added together. This can easily accomplishedI23,. . etc., are defined in the same way as I,,. by ensuring that any two adjacent values of y,; have

Substituting eqns (9) and (11) into eqn (5) gives the same direction.

3.2. Element sti@ess matr ix formulat ion

The variational indicator IT of Mindlin platetheory, according to [5], in linear elastic static analysisis

l-I=;

s

c;L&dA +;

s

~,:L?&,~ A - w> (14)

A A

where W is the work done by the applied loads.Using the displacement functions of eqn (I 3) and

the strain-displacements relations given in Section 2into eqn (14) gives

l-l = ~{S~}rpc,]{S~} - {S } {f J, (15)

4. NUMERICAL EXAMPLES

4.1. Eigen anal ysis of t he sti ffness mat ri x

From the eigen analysis of the element stiffnessmatrix it was found that there are three zero eigen-values. This shows that this element can present thethree rigid body motions for plate bending.

4.2. Locking test

Four elements were used to model a quartersimply supported plate under uniform load. Thefull side length of the plate, a was taken to be 10.0while the thickness, h, was varied. Satisfactory sol-utions were obtained for cases with a/h ratio up to106.


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Fig. 3. Assembling any adjacent elements (y,?, must have thesame direction).

4.3. Conv ergence t est s

4.3.1. Squar e and rect angul ar plat es. In this section

convergence studies were carried out on uniformlyloaded plates with different boundary conditions. Asymmetric quadrant plate was divided into a meshconsisting of N x N elements. Poissons ratio v andthe shear factor were taken as 0.3 and 0.833,respectively. The moments in Tables l-6 are calcu-lated directly at the specified points without anysmoothing. The following cases were considered.

(a) Simply supported square plate. Table 1 showsthe results for the central deflection, the bending atthe centre and the shear at the middle-edge of asquare plate with two different h/a ratios of 0.1 and0.01.

(b) Clamped square plate. The results of conver-gence tests for a uniformly loaded clamped squareplates with two different ratios for h/a (=O.l, 0.01)are presented in Table 2. The results are generally ingood agreement with those of [14].

(c) Simply supported rectangular plate. The cen-tral deflection and the central bending moment of arectangular plate with the following aspect ratiob:a:h=2:1:(0.1andO.Ol)aregiveninTable3.Theexact solution for this case using Reissner platetheory is available in [15].

(d) Square plate with two opposite edges simply

supported and the other two edges clamped. Table 4shows convergence study for this case. The thick-ness/length ratio (h/a) was taken to be 0.02. Theresults are in good agreement with those of [16].

(e) Square plate with two opposite edges simplysupported and the other two edges free. The resultsof convergence tests for this case are given in Table 5.They are very good in comparison with those of [ 161.

Table 1. Uniformly loaded, simply supported square plate

Central displacement Central moment

Meshdivision h/a = 0.1 hlo = 0.01 h/a =O .l h/a = 0.01

2x2 0.448 0.426 0.500 0.5013x3 0.436 0.415 0.489 0.4894x4 0.432 0.411 0.484 0.4855x5 0.43 I 0.409 0.482 0.482

Ref. [14] 0.427 0.406 0.479 0.48 1

Multiplier IO -q /D IO -*qa4/D IO -qa2 10mqa2

Meshdivision

2x23x34x45x5

Ref. [14]

Multiplier

Table 2. Uniformly loaded, clamped square plate


h/a = 0.1 h/a = 0.01 h/a =O .l h/a = 0.01

0.1372 0.1151 0.2238 0.23870.1449 0.1214 0.2317 0.23160.1473 0.1237 0.2315 0.23020.1485 0.1248 0.2317 0.2298

0.1500 0.1265 0.2310 0.2310

10 -qa4/D IO -q /D IO -qa I O- W

Table 3. Uniformly loaded, simply supported rectangular plate (6/a = 2)


Meshdivision h/a = 0.1 h/a = 0.01 h/a = 0.1 h/a = 0.01

2x2 0.1203 0.1166 0.1048 0.10503x3 0.1177 0.1131 0.1037 0.10294x4 0.1154 0.1120 0.1023 0.10235x5 0.1150 0.1115 0.1021 0.1021

Ref. [5] 0.1 136 0.1106 0.1189 0.1189

Multiplier IO -q /Eh) IO -qa4/Eh IO-qa lo-qa


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876 0. L. Roufaeil

Table 4. Uniformly loaded square plate with two oppositeedges simply supported (along J? = 0, a) and the other two

where Y is the distance between any point and the

edges clamped (along x = 0, a) centre and a is the radius of the plate.Note that in [17] the value of the shear factor K in

w centre: M, centre: M, centre:Mesh &ID qa qa

the above equation was taken as 213. In this work K*is 0.833. A comparison between the exact solution

3x3 0.001856 0.03304 0.023994x4 0.001890 0.03314 0.02419

and the present element using 12 and 48 elements is

5x5 0.001904 0.033 18 0.02429 given in Fig. 5 for a/h = 50, 5, 2.5. The results are in

Ref. [6] 0.001930 0.03320 0.02440excellent agreement with the exact solution of [17].Also, results for different a/h ratios of a uniformly

Table 5. Uniformly loaded square plate with two oppositeedges simply supported (along J: = 0, a) and the other two

edges free (along x = 0. a)

w centre: M ree edge: M, centre: M, freeMesh qa4/D &/D qa edge: qa

3x3 0.0127 0.1447 0.0264 0.12794x4 0.0129 0.1472 0.0266 0.1300

5x5 0.0130 0.1482 0.0268 0.1308Ref. [6] 0.0131 0.1500 0.0268 0.1300

4.3.2. Circular plates. Three different meshes areshown in Fig. 4. The number of elements (Nel) are3, 12 and 48. The exact solution for the displacementsalong the radius according to [17] is

loaded clamped plates using 3, 12 and 48 elements aregiven in Table 6. The results in Table 6 and in Fig.5 are given as a nondimensional parameterG = 64w D/ qa4.

4.3.3. Skew pl at es. Figure 6 shows a 60 skew platewith two opposite edges simply supported and theother two edges free. The properties for this plate areh =O.l, L = 100 and v =0.31. Razzaque [18] hasused a mesh of triangular elements to analyse thisproblem. Table 7 shows the results for this problemwith different meshes. The moment M, at the centrewas taken as the average of the moments for the foursurrounding elements. The results are in very goodagreement with those of [18].

4.4. EJSect o esh distort ion

(a*-r*)*+ 3n2y;i(a* r*) , (16) In this section the influence of mesh distortionon the solution accuracy is investigated. The

(a)

(b)

Fig. 4. Mesh division for a circular plate. (a) Nel = 3. (b) Nel = 12. (c) Nel = 48.


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I.2 r1.0 I-.

- Exact0.8 -

13 0.6 5 Liz :;:

0.4 - \

0.2 - a/h = 50 \,

I I I I I I I [ ._A

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 I.0

0))da

1.4 7

.2 -y*\*

I.0 -

0.8 -I3

l\,

0.6 - \

0.4 -0.2 - a/h = 5

A*,

I I I I I I I.

d0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

(c)r/a

2.01.8I.6I .41.2

13 I.00.80.60.40.2

r-

- a/h = 2.5

I I I I I I0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 I.0

r/a

Fig. 5. Displacements along the radius of a uniformly loaded circular plate.

Table 6. Uniformly loaded, circular clamped plate

Mesh division

0th 3 elements 12 elements 48 elements

Central displacement

Exact [17] Multiplier

50 I 0750:.5 2.3646.4673

1.0122 1 SW861.2378.8019 1.1921.7419

Central moment

1.00181.1828.7314 qa4/64D

50 0.872 0.804 0.8188 0.81255 I.060 0.974 0.8144 0.8125 IO-&2.5 1.374 1.002 0.8152 0.8125


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+---L= loo-----~

Fig. 6. Skew plate, simply supported along two sides andfree on the others.

Table 7. Uniformly loaded skew plate with

problem of a uniformly loaded simply supportedplate is modelled using the two different meshesshown in Fig. 7(a) and (b). For the mesh in Fig. 7(a)the value

-11_=0.998 and 0.997%1,,,,,

for h/L = 0 1 and 0.01, respectively.With reference to Fig. 7(b), the mesh distortion

parameter (LIP) is defined as DP = 2a/L The per-centage errors for different distortion parameters forh/L =O l and 0.01 are given in Fig. 8. This figure

two opposite edges simply supported and. rthe other two eages tree

Central displacement M, centre

Present work Ref. [18] Present work Ref. [IX]Mesh

Division

2x2 0.712 0.9104x4 0.893 0.97 I8x8 0.967

Multiplier O O07945qL4/D

0.623 0.7930.870 0.9560.964

O O9589qL?

b)

(a)Q I- a&L/2-a- I-.

Fig. 7. Modelling simply supported plate using two different distorted meshes.

10

5

O

E

5

e . h/L=O.l

h/L = 0.01

DP

Fig. 8. The percentage errors for different values of distortion parameters (DP).


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shows that this element behaves very well even forquite distorted elements.

5 DlSCUSSlON AND CONCLUSION

A four-node quadrilateral plate bending elementfor thin and moderately thick plates based on

Mindlin plate theory is presented in this work. These

elements neither have the problem of shear locking

nor exhibit any mechanism when an appropriateorder of numerical integration is used. Convergencetests showed that this element gives very good resultsfor both thick and thin plates.

REFERENCES

I. R. D. Mindlin, Influence of rotary inertia and shear onflexural motions of isotropic elastic plates. J. Appl.Mech., ASME 18, 31-35 (1951).

2. E. Reissner, The effect of transverse shear deformationon the bending of elastic plates. J. Appl. Mech., ASME12, 69-77 (1945).

3. 0. C. Zienkiewicz. The Finite Element Method.McGraw-Hill. New York (1977).

4. K. J. Bathe, Finite Elemenf Procedure in EngineeringAnalysis. Prentice-Hail, Englewood Cliffs, NJ (1982).

5. T. J. R. Hughes, The Finite Element Method. Prentice-Hall, Englewood Cliffs, NJ (1987).

6. M. M. Hrabok and T. M. Hrudey, A review andcatalogue of plate bending elements. Compur. Srruct. 19,

479-495 (1984).

8.

9.

IO.

II.

12.

13.

14.

15.

16.

17.

18.

S. Ahmad, B. M. Irons and 0. C. Zienkiewicz, Analysisof thick and thin shell structures by curved elements.ht. J. Nurser. Mefh. Engng 2, 419451 (1970).0. C. Zienkiewicz, J. Too and R. L. Taylor, Reducedintegration technique in general analysis of plates andshells. Inr. J. Numer. Merh. Engng 3, 2755290 (1971).S. F. Pawsay and R. W. Clough, Improved numericalintegration of thick shell finite elements. Inf. J. Numer.Me&. Engng 3, 545-586 (1971).A. C. Cassell and R. E. Hobbs. Dvnamic relaxation,high speed computing of elastic. structures. Proc.IUTAM Symp. Liege (1971).0. L. Roufaeil and D. J. Dawe, Vibration analysisof rectangular Mindlin plates by finite strip method.Cornput. Sfrucf. 12, 8333842 (1980).D. J. Dawe and 0. L. Roufaeil, Rayleigh-Ritz vibrationanalysis of Mindlin plates. J. Sound Vibr. 69, 345-359(1980).0. L. Roufaeil, New rectangular plate bending elementsand their performance. Compur Struct. 50(l). 59-65(1994).

E. Hinton and H. Huang, A family of quadrilateralMindlin plate elements with substitute shear strainfields. Compur. Strucr. 23, 409431 (1986).V. L. Salerno and M. A. Goldberg, Effect of sheardeformations on the bending of rectangular plates.J. Appl. Mech., ASME, 54-58 (1960).J. Jirousek and A. Bouberguig, A contribution toevaluation of shear forces and reactions to Mindlinplates by using isoparametric elements. Compur. Sfrucl.19, 793-800 (1984).S. Timoshenko and S. Woinowsky-Krieger, Theor ofPlates and Shells. McGraw-Hill, London (1959).A. Razzaque, Program for triangular bending elementswith derivative smoothing. Int. J. Numer. Meth. Engng

6, 333-343 (1973).

Four Node Plate Bending Element

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