A Shear Locking Free Six-node Mindlin Plate Bending Element

8/10/2019 A Shear Locking Free Six-node Mindlin Plate Bending Element

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548

M. H. VERWOERDnd A. W. M. KOK

Fig. 1. Rigid pin and midplane deformation.

The variables E and v are the Youngs modulus

and Poissons ratio, and k is the shape factor

k = 5/6).

Finite element formulation

For the six-node triangular element the same set of

shape functions [8] is used to approximate all the

midplane displacements.

The displacements, the translations (U , v , w ) and

the rotations (4, and 4,) are related to the nodal

displacements II: = [up, up, wp, c ,, &] by shape

functions a, x, y). The Appendix gives the expression

for the shape functions of the quadratic triangular

plate element.

The strain vectors L, and L, of eqns (2) and (3) are

now described by:

or

c, = B,u

(5b)

and

or

L, =

B,u

(6b)

Strain energy

The strain energy of the plate may be written as

the sum of membrane and transverse shear

energy.

aE,= +

s

aL;.a,dV+

s

ac:.a,dV.

7)

v

Substituting eqns (5b) and (6b) for &, and &,, the

strain energy is discretized for an element, and given

by,

+

s

duB;GB,udV. (8)

For thick plates the usual three-point integration

rule yields acceptable results. For thin plates, trans-

verse shear deformations dominate the stiffness

matrix (shear locking).

The base of the poor behaviour is found in the

inhomogeneous composition of the contributing

terms to the shear strains. The strains in the F.E.M.

are calculated following the kinematic relations of

eqn (3).

Shear locking is assumed to be introduced by the

obligate quadratic terms of y, vs the linear approxi-

mation of w,,. The result is an overly stiff element.

A refined transverse shear strain approximation is

proposed by the addition of correction terms to the

displacement field w. The function of these correction

terms is to neutralize the quadratic contribution of

f& in yXZ s f#~, n yyZ.

Correction terms

Shear strains are obtained by:

7x1= 4. + w,,

@a)

Yyl= - 4, + w

Y

For each direction s, the transverse shear strains can

be written as

Y,= 9, + w.,,

(9b)

where 4 is the quadratic polynomial and w,, the

linear polynomial. Shear locking is introduced by

the quadratic terms of 4,. The shear locking of the

SHELL6 element can be eliminated by addition of

correction terms, Aw.

or

rt=r,+Av,

(loa)

Y: = w,, + 4. + Aw,,

(1W

The correction term Aw should be taken in such a

way that rj

will be linear with respect to every

direction s.

The condition of the linear transverse shear strain

requires that

Y

* CO

s, .7

for every direction s.

(11)


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A shear locking free six-node Mindlin plate bending element

549

Using the kinematic relations of eqn (9b) the

condition to the displacement field is now

where

(12a)

Aw., = Aw,,, cos a + 3Aw,, cos* a sin

a

~Aw,,,,~ os

a

sin*

a

Aw,, sin a

(12b)

d.,, = &,XX os a + (2#5,, - 9,,) cos* a sin a

+ (& YY

21$,,) cos a sin* a - 4r.yy in a

(12c)

for each angle

a.

To this condition eqn (12a) is satisfied if:

The solution Aw is now given by:

Aw = - i (x*a:, + Zxya:, + y*a;J

x (x4,, - y ) - aAw*

(14a)

with discrete values

Awi+ =

-b(xfa:, + 2xiyia~Y+y~a~,,)(xi~Y-yi~X)

or

(14b)

The Appendix gives the expression for a,, aYY, Xv.

The refined shear strains in the x- and y-directions

are now given by:

in which the correction terms Ay, and AyYZare,

respectively,

by,, = Aw,, =

(f

xya: + iy2a:, - a:P,)

-(~x2a:X+fxya:Y+~y2a;Y+a:P2)~Y

(15b)

and

Ar,, = Aw,, = (ix2aiX + f xya: +fy2a;,, - a;P,)

-(ix2a:,+fxya6+a;P2)qbY. (1%)

The shear strains y$ and y; are

linearly with respect to x and y.

now distributed

NUMERICAL INTEGRATION

RULES

For both the SHELL6 element

SHELL6* element the three-point

has been applied.

and the refined

integration rule

Numerical test

A symmetric quadrant of a uniformly loaded,

simply supported, square plate is idealized by 32

elements for a thin plate of span/thickness ratio

L/t =

50.

To compare the quality of the refined

SHELL6* element the same quadrant has been ideal-

ized by 16 eight-node plate elements (SHELLS) too.

The exact solution for this class of problems is given

in [9].

L=lOrn

t=0 2m

E=lON/m

x

G lON/m

VSO

Fig. 3. Simply supported square plate under uniformly

distributed load.

ig. 2. Correction terms A.w.


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55

M. H. VERWOERD and A. W. M. KOK

601

I

I

2

L

0

2

-x [ml

-+x hl

Fig. 4. Moments m?,. and PI,, along line of symmetry y = 0.

601

50 -

30 -

SHELL6*/SHELLB

----x

[ml

-60

SHELL 6

-60

i

I

I

I

1 I

0

2 L

-x [ml

Fig. 5. Shear forces

qy

and

q_

along line of symmetry y = 0

RESULTS REFERENCES

The computed displacements of SHELL6 and

SHELL6* are very close to the exact results and are

not shown here. Results of the moments and the

shear forces are shown in Figs 4 and 5.

1.

2.

3.

4.

5.

6.

I.

8.

9.

I. Fried, Shear in C? and C bending finite elements. Inr.

J. Solids Struct. 9, 449 (1973).

0. C. Zienkiewicz, R. L. Taylor and J. M. Too,

Reduced integration techniques in general analysis of

plates and shells.

Int. J. Numer. Meth.

Engng 3,275-290

(1971).

T. J. R. Hughes, M. Cohen and M. Haoun, Reduced

and selective integration techniques in the finite element

analysis of plates. Nucl. Engng Des. 46, 203 (1978).

0. C. Zienkiewicz and E. Hinton, Reduced integration,

function smoothing and non-conformity in finite ele-

ment analysis (with special reference to thick plates).

J. Franklin Inst. 302, 443461 1976).

R. D. Mindlin, Influence of rotary inertia and shear on

flexural motions of isotropic plates.

J. uppl. Mech.

18

31-38 (1951).

T. J. R. Hughes and T. E. Tezduyar, Finite elements

based upon Mindlin plate theory with particular refer-

ence to the four-node bilinear isoparametric element.

J. appl. Mech ASME 48, 587-596 1981).

E. Reissner, The effect of transverse shear deformation

on the bending of elastic plates.

J. appt. Mech., Trans.

ASME 12, A69-A77 (1945).

C. S. Desai and J. F. Abel,

Introduction to the Finite

Element Method. Van Nostrand Reinhold, New York

(1972).

S. Timoshenko and S. Woinowky-Kreiger, Theory of

Plates and Shells,

2nd Edn. McGraw-Hill, New York

(1940).

CONCLUSIONS

The introduction of transverse shear strain in

the family of Mindlin element leads to shear

locking problems with decreasing thickness of

triangular and quadrilateral plate elements. Re-

duced or selective integration techniques can

overcome shear locking for four-node and eight-

node

elements. For triangles, reduced inte-

gration techniques do not solve the shear locking

problem.

For triangular plate elements shear locking can be

effectively eliminated by the introduction of the cor-

rection terms, Aw. These correction terms neutralize

the inhomogeneous quadratic terms in the shear

strain and successively the shear locking effects, This

new element leads to much more accurate results for

moments and shear forces.


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A shear locking free six-node Mindlin plate bending element

551

APPENDIX

The functions qxx, qYYand aLxyare the components of

the vectors a,, aYY nd axy, respectively,

The expression of the shape functions a,@, y) in vector a

are

L,(2L, - 1)

L&L, - 1)

4

0

0

a= L,(ZL,-1)

0

4

0

4LI L*

4

4

4

4L, L3

a,, =

%v =

0

0

axY=

4

4&L, _

-8

0

-4

whereL,=x;L,=y;L,=l-x-y;O

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A Shear Locking Free Six-node Mindlin Plate Bending Element