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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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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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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