- Email: [email protected]
Two generalized conforming plate elements based on semiloof constraints
Cmjmters d S~ructurecs Vol. 47, No. 2, pi. FTinted in Great Britain.
299-304,
1993
0043-7949/93 %.a? + 0.00 1993 Pq,amon Rear Ltd
0
TWO GENERALIZED CONFORMING PLATE ELEMENTS BASED ON SEMILOOF CONST~I~S LONG ZIIIPRI Department of Civil Engineering, Northern Jiaotong University, Beijing 100044, People’s Republic of China
Abstract-In this paper, the merits of both the generalized conforming element and the semiLoof element are utilized to establish nine-DOF triangular and 12DOF rectangular thin-plate bending elements. The element DOFs are defined with the conventional displacements at comer nodes and the semiL.oof constraints are used in the formulation. By using the concept of a generalized conforming element, these elements can pass Irons’ patch test and accurate results are obtained with a much lower computational effort.
1. INTRODUCIlON
A great deal of effort has been spent constructing plate bending elements [l, 21. For the case of nineDOF triangular plate elements, it is concluded in [I] that the most efficient elements are the DKT (discrete Kirchhoff theory) and HSM (hybrid stress model) elements. The generalized conforming element developed in [3,4] appears to be another simple, efficient and reliable element for thin-plate analysis. In [5,6], Irons developed a complex, but attractive element, the semiLoof element. Later, Pian and Sumihara [7’J formulated the hybrid semiLoof element. In this paper, by combining the concept of the generalized conforming element and the semiLoof element nine-DOF triangular and 12-DOF rectangular plate bending elements are formulated.
In order to solve (A) in terms of {q}: it is necessary to establish 12 compatibility conditions. First, apply the compatibility conditions for deflections at the comer nodes (w-9),=0
E,,=w,,
&=wz
(w -r3)j=0
1 ,=~t~,,-~,,,-%(JI,3-~~,,
II 6 =
rm~j,
(5)
conditions
for de&c-
(j--4,5,6)
(6)
Ad, 1,) 2, can be solved
A triangular thin plate element with nine-DOF is shown in Fig. 1. The vector of nodal DOF {q}# is defined as
w=
&=ws.
Then, apply the compatibility tion at the mid-side nodes
14=~(~~2-~~3)-$,h,-h,,
where wi, J/Xiand lLyidenote deflection and rotations at node i (i = 1,2,3). Along each side of the element, the deflection $ is assumed to be cubic and the normal slope $,, linearly distributed. The element deflection field w is assumed to be a polynomial with 12 terms, expressible in terms of area coordinates L,, L2 and L, as follows:
(4)
Izl, A,, 1, can be solved
2. TPIANGULAR ELEMENT ISL
(1)
(i=1,2,3)
4
(#xl
-
+x*1
-
2
CJ/yl
-
(7)
#yt)*
whereb,=y,-y,,c,=x3-x2.andsoon. Finally, apply the compatibility conditions for normal slopes at Gauss points A, and Bk along each side
($-&),=O,($$m)sk=O, (k-1,2,3) (8)
(2) A,, . . . , Al2 can be solved. In this solution, the last
where
three
coeflkients
are
equal,
i.e.
& = l,, = &2.
{A} = [A, A* . . . &Jr VI
=
IL,
Lz
L3
w3
J%
J%L*
L*L,(L,-L,)
L,w53-LL,)
Jw,(L,
-LA
JqL,L,
GL,L,
L:L,LJ. (3)
299
300
LONG
ZHIFEI
Therefore, the element deflection field can be rewritten in terms of 10 coefficients w = FIP),
(9)
where {X}= [A,& FJ=r-b Combining
. . . 1,,] L,
L,
L2-G L,L,
L,L,
L*L3(L,-L,)
L3L,(L,--Ll)
L,L*G-L*)
L,L*L,l.
(10)
the above results, (1) can be expressed in terms of {q}’ as follows:
(11) where
[Al = [[A,1L&l L&II
[A,]= fk2
1
0
0
0
0
0
0
0
0
0
0
0
0
-I‘c
0
+j
+
r3)
-f(r3+3) t(3 -r2) r3 - r2
[A21=
(12)
h(C,
-
2
$2
3r2c2 + 3r3c,)
i(-3r,c,-c3+8c,) +(3r,c,
(134
-I‘b 3 +(b,
- 3r2b2 + 3r3b3)
2(-3r3b3-b3+8b2)
- c, + 8c,)
f(r2 c2 + r3c3)
%(3r,b,-b2+8b3) -i(r,b,
+ r3b3)
0
0
0
1
0
0
0
0
0 -i‘b 1
0
2,
0
0
0
-i‘C 3 h(3r3 c) -
$(3-r,)
t(r3 + r,) +(r,+3)
0
~3 +
8~3)
G(3r3b3 -b, *(b,-3r3b3
A(-3r,c,-c,+8c,)
$(-3r,b,-b,+86,)
+ 3r,b,)
$(r3b3+rlbl) _
f(r3c3 + rI CI 1
0
0
0
0
0
0
0
0
0
?C,
$1
0
tC2
0
0
i(3 - r,) i(r, + r2) i-2 -
rl
+ 8b,)
i$(c, - 3r3c3 + 3r, c,)
6 - r3
-t(r2+3)
U3b)
fb3
(13c)
h( - 3r2c2 - c2 + 8c,) h(3r, ~1- CI+ A(c,-3r,c,
8~2)
+ 3r2c2)
f(r,c, + r2c2)
3(-3r2b2-b2+8b,) *(3r, +(b,
b, - b, + 86,) - 3r, b, + 3r2b2)
?(r,b,
+ r2b2)
!
301
Two generalized conforming plate elements
Fig. 1. Triangular element, 1, 2, 3 are corner nodes; 4, 5, 6 are mid-size nodes; and A,, Bk are two Gauss points along Fig. 3. Pure bending patch test.
each side (k = 1, 2, 3). Area coordinates of A, and B, are
A,[,,f(1+s), ;(l -$=)I
The element deflection field w is assumed to be a polynomial with 16 coefficients
=(&+1*5
where d; - d:
d; - d: r2 =
r’=d:
+(5*-1)(4+&t
df - d; r’=d:
+(t1* - l)(& + M
Substituting eqn (11) into eqn (9), w can be expressed in terms of {q}’
w = rmAl{qY.
+
cr*- ‘h2
+&trl)
+
M5* - t12)l,
LSL-Rl2
where r =x/a coordinates. Apply the conditions
aw
(
-j+”
following
),=o ,
42etl>
(17)
r) = y/b
and
(w-@)~=O
+ A,,rt +
- l)W,, + A,.$(+ I,sq
(15)
The element stiffness matrix can be derived by a conventional procedure. This element is called LSLT9. Note that the LT element in [8] derived with the integral compatibility conditions is equivalent to the present element. However, the formulation procedure in this paper based on the discrete compatibility conditions appears to be simpler. ELEMENT
++I
(14)
yg--
3. RECTANGULAR
+&tl +&&I)
are
dimensionless
discrete
compatibility
(i=1,2,...,8)
V=C,,D
I,...,
C,,D,) (18)
can be solved in terms of {q}‘. In this solution, the last four coefficients are zero {A}
A rectangular thin-plate element with 12 DOF is shown in Fig. 2. Along each side of the element there are two Gauss points Ci and Di (i = 1, 2, 3, 4). The vector of nodal DOF {q}’ is defined as
1,s = A,, = 11, = 11, = 0
(19)
and the first 12 coefficients can be expressed in terms {q}‘=
[w,
i,hx, I+$, . . .
w., i+bxs $,I?
(16)
where
Y. rl
4 a.
.6 -
i. (W,,V,,,
vyl;
=.
%__
__Cl
:
:
5
Of {q}’ as f”‘lows:
I ’
c
Table 1. The exact solution of the pure bending patch test problem aW
l
1)1.v-.~2~%v~2’
Fig. 2. Rectangular element.
VyI)
Node
2 3 4
W
16.4835 -35.1648 -41.5934
ax
3.2967 O.OOC@ 2.9070
aw ay
0.0000 -8.7912 - 10.9890
302
i?kWEl
hNG
200200 2
0
0
2
0
0
-2
0
0
2
0
0
-2
0
0
2
0
0
-2
0
Cl 0 2
-2
0
0
0
0
0
0
0
a
0
0
a
0
0
-a
0
a
b/3
-1
a
-b/3
-1
a
b/3
1
a
-b/3
a
0
0
-a
0
0
a
0
0
-a
0
-9a
-3b
3
-9a
36
9a
36
3
9a
-3b
8
8
2
8
%-
-3 ---
0 0
0 0
b -b
00
00
-b -b
00
00
b b
00
00
-b b
1
a/3
b
1
-a/3
b
-1
a/3
b
-1
-a/3
b
-3a
--96 8
3 5
-3a 8
9b
-3
3a
9b
3
3a
-%-
--2882_ii-
-U
[A,]=;
2
-3 i 28
-
Once {A) is solved, the element stiffness matrix can be derived by the conventional procedure. This element is called LSL-R12.
28828
8
1
(22)
-9b
-J 8
MXY= 2.5 kN. For the mesh shown in Fig. 4(b), the present element LSL-T9 reproduces the exact solution and passes the patch test.
4. NUMERICAL EXAMPLES Example element
1. Pure
bending
patch
test-triangular
Consider the pure bending patch test problem given in (91(Fig. 3). The two-el~ent patch with node 1 fixed is subjected to pure bending: MY= 1.0 kN. Assume v = 0.3, Eh3/12(1 -v*) = 1.0. The analytic solution for displacements is given in Table 1. The correct results are obtained with the LST-T9 element. Thus, the element passes the patch test. Example element
2. Pure
twisting patch
test-triangular
Consider the pure twisting patch test problem used in [lo] (Fig. 4). The analytic solution of plate in Fig. 4(a) under load P = 5 kN is M,= MY=Ot
Example 3. Square triangular element
plates
Simply
rapported
(b) Fig. 4. Pure twisting patch test.
uniform
load-
In [ 111, Jeyachandrabose and Kirkhope concluded that the HBS canon-Bergan-Syvertsen) f12] and CT (composite triangle) [13] elements are the two most accurate nine-DOF triangular thin-plate bending elements currently available. In order to compare the performance of the present element LSL-T9 with that of the CT element, the calculated results for central deI%ection and the moment of square plate under uniform load are given in Table 2 in which two mesh orientations, A and B 1131, are considered. The overall performance of LSL-TO is shown to be superior to that of CT.
/
w
under
Fig. 5. Rhombic plate.
303
Two generalized conforming plate elements Table 2. Square plate under uniform load-triangular
elements Clamped
Simply supported LSL-T9 Mesh (l/4 plate)
A
B
A
0.4024 (-0.9%)
0.39930 (-1.7%)
0.35118 (-13.6%)
0.12288 (-2.9%)
A
B
Central deflection 2x2 0.4014 (- 1.2%)
CT
LSGT9
CT
A
B
0.10768 (-14.9%)
0.14750 (16.6%)
0.10732 (- 15.2%) 0.12232 (-3.3%) 0.12468 (- 1.5%)
B
4x4
0.4051 (-0.3%)
0.4058 (-0.1%)
0.40439 (-0.5%)
0.39280 (-3.3%)
0.12544 (-0.8%)
0.12203 (-3.6%)
0.13221 (4.5%)
6x6
0.40574 (-0.1%)
(“E-g
(YE)
(YE)
(YYo)
(“lE)
0.12912 (2.0%)
0.12653 q1’/(lOOD)
0.406235 q1’/(lOOD) Central moment 2x2 0.5022 (4.9%)
0.5161 (7.8%)
0.49988 (4.4%)
0.43958 (-8.2%)
0.2909 (27%)
4x4
0.4798 (0.2%)
0.4917 (2.7%)
0.48347 (1.0%)
0.47005 (- 1.8%)
0.2386 (4.2%)
0.2380 (3.9) 0.2343 (2.3%)
6x6
0.47821 (-0.1%)
0.48551 (1.4%)
0.48090 (0.4%)
0.47493 (-0.8%)
0.23277 (1.6%)
0.23155 (1.1%)
0.29510 (28.8%)
0.20527 (- 10.4%)
0.24671 (7.7%) 0.23751 (3.7%)
0.22389 (-2.3%) 0.22605 (- 1.3%)
0.22905(qP/lO)
0.47886(q12/10)
Table 3. Central deflection and moment of rhombic plate under uniform load
Central deflection Mesh LSL, A9
DOF
2x2 27 4x4 75 4x6 105 8x8 243 12 x 12 507 Finite difference P41
Central moment M,
LSL-T9
A9 [14]
Ref. [15]
LSL-T9
A9 [14]
Ref. [15]
0.7229 0.7717 0.7848 0.7870 0.7903
0.7230 0.7718 0.7850 -
0.8414 0.8111 0.8057
0.9837 0.9780 0.9776 0.9645 0.9620
0.7602 0.9172 0.9473
0.9761 0.9739 0.9688
0.7945 q1’/lOOD
0.9589 q12/10
Example 4. Rhombic plate under uniform loadtriangular element
overall performance of LSL-T9 element is shown to be superior to that of the elements in [14] and [15].
The rhombic plate under uniform load in Fig. 5 was analysed in [14] and [lS]. The results of central deflection and moment are given in Table 3. The
Example 5. Square plate rectangular element
Table 4. Square plate under uniform load-rectangular elements Simply supported Mesh (l/4 plate)
LSL-Rl2
Central deflection 2x2 0.40513 (-0.3%) 4x4
0.40616 (-0.02%)
8x8
0.40623 1 (-0.001%)
Central moment 2x2 0.51243 (7.0%) 4x4
0.48730 (1.8%)
0 .. 0
0.48998 (0.4%)
0x0
Clamped
ACM
LSL-RI2
ACM
0.3939 (-3.0%)
0.12265 (-3.1%)
0.4033 (-0.7%) 0.4056 (-0.2%)
0.12586 (-0.5%) 0.12644 (-0.08%)
0.1403 (11.0%) 0.1304 (4.0%)
0.52169 (8.9%) 0.48920 (2.2%) 0.48166 (0.6%)
0.25728 (12.3%) 0.23680 (3.4%) 0.23107 (0.9%)
under uniform load-
The ACM element is a well known and reasonably accurate rectangular plate element with 12 DOFs. The results for square plate under uniform load calculated with both elements LSL-Rl2 and ACM are compared in Table 4. Apparently the LSL-RI2 element gives more accurate results in all cases. 5. CONCLUSION
0.1275 (0.8%) 0.27783 (21.3%) 0.24050 (5.0%) 0.23191 (1.2%)
A new method to formulate thin-plate bending elements based on the concept of the generalized conforming element and the semiLoof element is proposed. The main points are: (1) the semiLoof constraints are used in the formulation, while the element DOFs are different from that of the semiLoof element; (2) the excellent performance of the generalized conforming element is preserved, while the integral compatibility conditions are discarded in the formulation. Based on this method, two thin-plate bending elements, LSL-T9 and LSL-R12, are constructed.
LONG ZHIFEI
304 These new elements can pass the patch reliable and accurate results with a effort. Numerical computational presented for these elements and other ements illustrate the excellent bchaviour elements.
test and give much lower comparisons popular elof these new
I. T. H. H. Pian and Sumihara, Hybrid scmiLoof elements
8.
9. REFERENCES
1. J. L. Batoz, K. J. Bathe and L. W. Ho, A study of three-node triangular plate bending elements. Inr. 1. Numer. Meth. Engng IS, 1771-1812 (1980). 2. M. M. Hrabok and T. M. Hrudey, A review and catalogue of plate bending finite elements. Compuf. Strucr. 19, 479-495 (1984). 3. Long Yuqiu and Xin Kegui, Generalized conforming element for bending and buckling analysis of plates. Finite Elements in Analysis and Design 5, IS-30 (1989). 4. Long Yuqiu and Zhao Junqing, A new generalized cont&ming triangular element for thin plates. Communs aool. Numer. Meth. 4. 781-792 (1988). 5. B.’ M. Irons and S: Ahmad,. Techniques of Finite Elements, Chap. 18. Ellis Horwood. Chichester (1980). 6. B. M. Irons, The semiLoof shell element. In Finite Elements for Thin Shells and Curved Members (Edited by D. G. Ashwell and R. H. Gallagher), pp. 197-222. John Wiley, London (1976).
10. 11.
12.
13.
14.
15.
for plates and shells based upon a modified Hu-Washizu principle. Cornput. Struct. 19, 165-173 (1984). Long Zhifei, Triangular and rectangular plate elements based on generalized compatibility conditions. Proc. Second World Congress on Computational Mechanics, Vol. 2, pp. 594-597. Stuttgart (1990). R. L. Tavlor. J. C. Simo. 0. C. Zienkiewicz and A. C. H. Chan, The patch test-a condition for assessing FEM convergence. Int. J. Numer. Meth. Engng 22, 39-62 (1986). A. Razzaque, The patch test for elements. Inr. J. Numer. Meth. Engng 22, 63-71 (1986). C. Jeyachandrabose and J. Kirkhope, Construction of new efficient three-mode triangular thin plate bending elements. Compuf. Strucr. 23, 587-603 (1986). L. Hansen, P. G. Bergan and T. G. Syvertsen, Stiffness derivation based on element convergence requirements. In The Mathematics of Finite Elements and Applications (Edited by J. R. Whiteman), pp. 83-96. Academic Press, London (1979). A. J. Fricker, An improved three-noded triangular element for plate bending. Im. J. Numer. Meth. Engng 21, 1055114 (1985). A. Razzaque, Program for triangular bending element with derivative smoothing. Inr. J. Mefh. Engng 6, 333-343 (1973). 0. C. Zienkiewicz and D. Lefebvre, A robust triangular plate bending element of the Reissner-Mindlin type. Inr. J. Numer. Mesh. Engng 26, 1169-1184 (1988).
299-304,
1993
0043-7949/93 %.a? + 0.00 1993 Pq,amon Rear Ltd
0
TWO GENERALIZED CONFORMING PLATE ELEMENTS BASED ON SEMILOOF CONST~I~S LONG ZIIIPRI Department of Civil Engineering, Northern Jiaotong University, Beijing 100044, People’s Republic of China
Abstract-In this paper, the merits of both the generalized conforming element and the semiLoof element are utilized to establish nine-DOF triangular and 12DOF rectangular thin-plate bending elements. The element DOFs are defined with the conventional displacements at comer nodes and the semiL.oof constraints are used in the formulation. By using the concept of a generalized conforming element, these elements can pass Irons’ patch test and accurate results are obtained with a much lower computational effort.
1. INTRODUCIlON
A great deal of effort has been spent constructing plate bending elements [l, 21. For the case of nineDOF triangular plate elements, it is concluded in [I] that the most efficient elements are the DKT (discrete Kirchhoff theory) and HSM (hybrid stress model) elements. The generalized conforming element developed in [3,4] appears to be another simple, efficient and reliable element for thin-plate analysis. In [5,6], Irons developed a complex, but attractive element, the semiLoof element. Later, Pian and Sumihara [7’J formulated the hybrid semiLoof element. In this paper, by combining the concept of the generalized conforming element and the semiLoof element nine-DOF triangular and 12-DOF rectangular plate bending elements are formulated.
In order to solve (A) in terms of {q}: it is necessary to establish 12 compatibility conditions. First, apply the compatibility conditions for deflections at the comer nodes (w-9),=0
E,,=w,,
&=wz
(w -r3)j=0
1 ,=~t~,,-~,,,-%(JI,3-~~,,
II 6 =
rm~j,
(5)
conditions
for de&c-
(j--4,5,6)
(6)
Ad, 1,) 2, can be solved
A triangular thin plate element with nine-DOF is shown in Fig. 1. The vector of nodal DOF {q}# is defined as
w=
&=ws.
Then, apply the compatibility tion at the mid-side nodes
14=~(~~2-~~3)-$,h,-h,,
where wi, J/Xiand lLyidenote deflection and rotations at node i (i = 1,2,3). Along each side of the element, the deflection $ is assumed to be cubic and the normal slope $,, linearly distributed. The element deflection field w is assumed to be a polynomial with 12 terms, expressible in terms of area coordinates L,, L2 and L, as follows:
(4)
Izl, A,, 1, can be solved
2. TPIANGULAR ELEMENT ISL
(1)
(i=1,2,3)
4
(#xl
-
+x*1
-
2
CJ/yl
-
(7)
#yt)*
whereb,=y,-y,,c,=x3-x2.andsoon. Finally, apply the compatibility conditions for normal slopes at Gauss points A, and Bk along each side
($-&),=O,($$m)sk=O, (k-1,2,3) (8)
(2) A,, . . . , Al2 can be solved. In this solution, the last
where
three
coeflkients
are
equal,
i.e.
& = l,, = &2.
{A} = [A, A* . . . &Jr VI
=
IL,
Lz
L3
w3
J%
J%L*
L*L,(L,-L,)
L,w53-LL,)
Jw,(L,
-LA
JqL,L,
GL,L,
L:L,LJ. (3)
299
300
LONG
ZHIFEI
Therefore, the element deflection field can be rewritten in terms of 10 coefficients w = FIP),
(9)
where {X}= [A,& FJ=r-b Combining
. . . 1,,] L,
L,
L2-G L,L,
L,L,
L*L3(L,-L,)
L3L,(L,--Ll)
L,L*G-L*)
L,L*L,l.
(10)
the above results, (1) can be expressed in terms of {q}’ as follows:
(11) where
[Al = [[A,1L&l L&II
[A,]= fk2
1
0
0
0
0
0
0
0
0
0
0
0
0
-I‘c
0
+j
+
r3)
-f(r3+3) t(3 -r2) r3 - r2
[A21=
(12)
h(C,
-
2
$2
3r2c2 + 3r3c,)
i(-3r,c,-c3+8c,) +(3r,c,
(134
-I‘b 3 +(b,
- 3r2b2 + 3r3b3)
2(-3r3b3-b3+8b2)
- c, + 8c,)
f(r2 c2 + r3c3)
%(3r,b,-b2+8b3) -i(r,b,
+ r3b3)
0
0
0
1
0
0
0
0
0 -i‘b 1
0
2,
0
0
0
-i‘C 3 h(3r3 c) -
$(3-r,)
t(r3 + r,) +(r,+3)
0
~3 +
8~3)
G(3r3b3 -b, *(b,-3r3b3
A(-3r,c,-c,+8c,)
$(-3r,b,-b,+86,)
+ 3r,b,)
$(r3b3+rlbl) _
f(r3c3 + rI CI 1
0
0
0
0
0
0
0
0
0
?C,
$1
0
tC2
0
0
i(3 - r,) i(r, + r2) i-2 -
rl
+ 8b,)
i$(c, - 3r3c3 + 3r, c,)
6 - r3
-t(r2+3)
U3b)
fb3
(13c)
h( - 3r2c2 - c2 + 8c,) h(3r, ~1- CI+ A(c,-3r,c,
8~2)
+ 3r2c2)
f(r,c, + r2c2)
3(-3r2b2-b2+8b,) *(3r, +(b,
b, - b, + 86,) - 3r, b, + 3r2b2)
?(r,b,
+ r2b2)
!
301
Two generalized conforming plate elements
Fig. 1. Triangular element, 1, 2, 3 are corner nodes; 4, 5, 6 are mid-size nodes; and A,, Bk are two Gauss points along Fig. 3. Pure bending patch test.
each side (k = 1, 2, 3). Area coordinates of A, and B, are
A,[,,f(1+s), ;(l -$=)I
The element deflection field w is assumed to be a polynomial with 16 coefficients
=(&+1*5
where d; - d:
d; - d: r2 =
r’=d:
+(5*-1)(4+&t
df - d; r’=d:
+(t1* - l)(& + M
Substituting eqn (11) into eqn (9), w can be expressed in terms of {q}’
w = rmAl{qY.
+
cr*- ‘h2
+&trl)
+
M5* - t12)l,
LSL-Rl2
where r =x/a coordinates. Apply the conditions
aw
(
-j+”
following
),=o ,
42etl>
(17)
r) = y/b
and
(w-@)~=O
+ A,,rt +
- l)W,, + A,.$(+ I,sq
(15)
The element stiffness matrix can be derived by a conventional procedure. This element is called LSLT9. Note that the LT element in [8] derived with the integral compatibility conditions is equivalent to the present element. However, the formulation procedure in this paper based on the discrete compatibility conditions appears to be simpler. ELEMENT
++I
(14)
yg--
3. RECTANGULAR
+&tl +&&I)
are
dimensionless
discrete
compatibility
(i=1,2,...,8)
V=C,,D
I,...,
C,,D,) (18)
can be solved in terms of {q}‘. In this solution, the last four coefficients are zero {A}
A rectangular thin-plate element with 12 DOF is shown in Fig. 2. Along each side of the element there are two Gauss points Ci and Di (i = 1, 2, 3, 4). The vector of nodal DOF {q}’ is defined as
1,s = A,, = 11, = 11, = 0
(19)
and the first 12 coefficients can be expressed in terms {q}‘=
[w,
i,hx, I+$, . . .
w., i+bxs $,I?
(16)
where
Y. rl
4 a.
.6 -
i. (W,,V,,,
vyl;
=.
%__
__Cl
:
:
5
Of {q}’ as f”‘lows:
I ’
c
Table 1. The exact solution of the pure bending patch test problem aW
l
1)1.v-.~2~%v~2’
Fig. 2. Rectangular element.
VyI)
Node
2 3 4
W
16.4835 -35.1648 -41.5934
ax
3.2967 O.OOC@ 2.9070
aw ay
0.0000 -8.7912 - 10.9890
302
i?kWEl
hNG
200200 2
0
0
2
0
0
-2
0
0
2
0
0
-2
0
0
2
0
0
-2
0
Cl 0 2
-2
0
0
0
0
0
0
0
a
0
0
a
0
0
-a
0
a
b/3
-1
a
-b/3
-1
a
b/3
1
a
-b/3
a
0
0
-a
0
0
a
0
0
-a
0
-9a
-3b
3
-9a
36
9a
36
3
9a
-3b
8
8
2
8
%-
-3 ---
0 0
0 0
b -b
00
00
-b -b
00
00
b b
00
00
-b b
1
a/3
b
1
-a/3
b
-1
a/3
b
-1
-a/3
b
-3a
--96 8
3 5
-3a 8
9b
-3
3a
9b
3
3a
-%-
--2882_ii-
-U
[A,]=;
2
-3 i 28
-
Once {A) is solved, the element stiffness matrix can be derived by the conventional procedure. This element is called LSL-R12.
28828
8
1
(22)
-9b
-J 8
MXY= 2.5 kN. For the mesh shown in Fig. 4(b), the present element LSL-T9 reproduces the exact solution and passes the patch test.
4. NUMERICAL EXAMPLES Example element
1. Pure
bending
patch
test-triangular
Consider the pure bending patch test problem given in (91(Fig. 3). The two-el~ent patch with node 1 fixed is subjected to pure bending: MY= 1.0 kN. Assume v = 0.3, Eh3/12(1 -v*) = 1.0. The analytic solution for displacements is given in Table 1. The correct results are obtained with the LST-T9 element. Thus, the element passes the patch test. Example element
2. Pure
twisting patch
test-triangular
Consider the pure twisting patch test problem used in [lo] (Fig. 4). The analytic solution of plate in Fig. 4(a) under load P = 5 kN is M,= MY=Ot
Example 3. Square triangular element
plates
Simply
rapported
(b) Fig. 4. Pure twisting patch test.
uniform
load-
In [ 111, Jeyachandrabose and Kirkhope concluded that the HBS canon-Bergan-Syvertsen) f12] and CT (composite triangle) [13] elements are the two most accurate nine-DOF triangular thin-plate bending elements currently available. In order to compare the performance of the present element LSL-T9 with that of the CT element, the calculated results for central deI%ection and the moment of square plate under uniform load are given in Table 2 in which two mesh orientations, A and B 1131, are considered. The overall performance of LSL-TO is shown to be superior to that of CT.
/
w
under
Fig. 5. Rhombic plate.
303
Two generalized conforming plate elements Table 2. Square plate under uniform load-triangular
elements Clamped
Simply supported LSL-T9 Mesh (l/4 plate)
A
B
A
0.4024 (-0.9%)
0.39930 (-1.7%)
0.35118 (-13.6%)
0.12288 (-2.9%)
A
B
Central deflection 2x2 0.4014 (- 1.2%)
CT
LSGT9
CT
A
B
0.10768 (-14.9%)
0.14750 (16.6%)
0.10732 (- 15.2%) 0.12232 (-3.3%) 0.12468 (- 1.5%)
B
4x4
0.4051 (-0.3%)
0.4058 (-0.1%)
0.40439 (-0.5%)
0.39280 (-3.3%)
0.12544 (-0.8%)
0.12203 (-3.6%)
0.13221 (4.5%)
6x6
0.40574 (-0.1%)
(“E-g
(YE)
(YE)
(YYo)
(“lE)
0.12912 (2.0%)
0.12653 q1’/(lOOD)
0.406235 q1’/(lOOD) Central moment 2x2 0.5022 (4.9%)
0.5161 (7.8%)
0.49988 (4.4%)
0.43958 (-8.2%)
0.2909 (27%)
4x4
0.4798 (0.2%)
0.4917 (2.7%)
0.48347 (1.0%)
0.47005 (- 1.8%)
0.2386 (4.2%)
0.2380 (3.9) 0.2343 (2.3%)
6x6
0.47821 (-0.1%)
0.48551 (1.4%)
0.48090 (0.4%)
0.47493 (-0.8%)
0.23277 (1.6%)
0.23155 (1.1%)
0.29510 (28.8%)
0.20527 (- 10.4%)
0.24671 (7.7%) 0.23751 (3.7%)
0.22389 (-2.3%) 0.22605 (- 1.3%)
0.22905(qP/lO)
0.47886(q12/10)
Table 3. Central deflection and moment of rhombic plate under uniform load
Central deflection Mesh LSL, A9
DOF
2x2 27 4x4 75 4x6 105 8x8 243 12 x 12 507 Finite difference P41
Central moment M,
LSL-T9
A9 [14]
Ref. [15]
LSL-T9
A9 [14]
Ref. [15]
0.7229 0.7717 0.7848 0.7870 0.7903
0.7230 0.7718 0.7850 -
0.8414 0.8111 0.8057
0.9837 0.9780 0.9776 0.9645 0.9620
0.7602 0.9172 0.9473
0.9761 0.9739 0.9688
0.7945 q1’/lOOD
0.9589 q12/10
Example 4. Rhombic plate under uniform loadtriangular element
overall performance of LSL-T9 element is shown to be superior to that of the elements in [14] and [15].
The rhombic plate under uniform load in Fig. 5 was analysed in [14] and [lS]. The results of central deflection and moment are given in Table 3. The
Example 5. Square plate rectangular element
Table 4. Square plate under uniform load-rectangular elements Simply supported Mesh (l/4 plate)
LSL-Rl2
Central deflection 2x2 0.40513 (-0.3%) 4x4
0.40616 (-0.02%)
8x8
0.40623 1 (-0.001%)
Central moment 2x2 0.51243 (7.0%) 4x4
0.48730 (1.8%)
0 .. 0
0.48998 (0.4%)
0x0
Clamped
ACM
LSL-RI2
ACM
0.3939 (-3.0%)
0.12265 (-3.1%)
0.4033 (-0.7%) 0.4056 (-0.2%)
0.12586 (-0.5%) 0.12644 (-0.08%)
0.1403 (11.0%) 0.1304 (4.0%)
0.52169 (8.9%) 0.48920 (2.2%) 0.48166 (0.6%)
0.25728 (12.3%) 0.23680 (3.4%) 0.23107 (0.9%)
under uniform load-
The ACM element is a well known and reasonably accurate rectangular plate element with 12 DOFs. The results for square plate under uniform load calculated with both elements LSL-Rl2 and ACM are compared in Table 4. Apparently the LSL-RI2 element gives more accurate results in all cases. 5. CONCLUSION
0.1275 (0.8%) 0.27783 (21.3%) 0.24050 (5.0%) 0.23191 (1.2%)
A new method to formulate thin-plate bending elements based on the concept of the generalized conforming element and the semiLoof element is proposed. The main points are: (1) the semiLoof constraints are used in the formulation, while the element DOFs are different from that of the semiLoof element; (2) the excellent performance of the generalized conforming element is preserved, while the integral compatibility conditions are discarded in the formulation. Based on this method, two thin-plate bending elements, LSL-T9 and LSL-R12, are constructed.
LONG ZHIFEI
304 These new elements can pass the patch reliable and accurate results with a effort. Numerical computational presented for these elements and other ements illustrate the excellent bchaviour elements.
test and give much lower comparisons popular elof these new
I. T. H. H. Pian and Sumihara, Hybrid scmiLoof elements
8.
9. REFERENCES
1. J. L. Batoz, K. J. Bathe and L. W. Ho, A study of three-node triangular plate bending elements. Inr. 1. Numer. Meth. Engng IS, 1771-1812 (1980). 2. M. M. Hrabok and T. M. Hrudey, A review and catalogue of plate bending finite elements. Compuf. Strucr. 19, 479-495 (1984). 3. Long Yuqiu and Xin Kegui, Generalized conforming element for bending and buckling analysis of plates. Finite Elements in Analysis and Design 5, IS-30 (1989). 4. Long Yuqiu and Zhao Junqing, A new generalized cont&ming triangular element for thin plates. Communs aool. Numer. Meth. 4. 781-792 (1988). 5. B.’ M. Irons and S: Ahmad,. Techniques of Finite Elements, Chap. 18. Ellis Horwood. Chichester (1980). 6. B. M. Irons, The semiLoof shell element. In Finite Elements for Thin Shells and Curved Members (Edited by D. G. Ashwell and R. H. Gallagher), pp. 197-222. John Wiley, London (1976).
10. 11.
12.
13.
14.
15.
for plates and shells based upon a modified Hu-Washizu principle. Cornput. Struct. 19, 165-173 (1984). Long Zhifei, Triangular and rectangular plate elements based on generalized compatibility conditions. Proc. Second World Congress on Computational Mechanics, Vol. 2, pp. 594-597. Stuttgart (1990). R. L. Tavlor. J. C. Simo. 0. C. Zienkiewicz and A. C. H. Chan, The patch test-a condition for assessing FEM convergence. Int. J. Numer. Meth. Engng 22, 39-62 (1986). A. Razzaque, The patch test for elements. Inr. J. Numer. Meth. Engng 22, 63-71 (1986). C. Jeyachandrabose and J. Kirkhope, Construction of new efficient three-mode triangular thin plate bending elements. Compuf. Strucr. 23, 587-603 (1986). L. Hansen, P. G. Bergan and T. G. Syvertsen, Stiffness derivation based on element convergence requirements. In The Mathematics of Finite Elements and Applications (Edited by J. R. Whiteman), pp. 83-96. Academic Press, London (1979). A. J. Fricker, An improved three-noded triangular element for plate bending. Im. J. Numer. Meth. Engng 21, 1055114 (1985). A. Razzaque, Program for triangular bending element with derivative smoothing. Inr. J. Mefh. Engng 6, 333-343 (1973). 0. C. Zienkiewicz and D. Lefebvre, A robust triangular plate bending element of the Reissner-Mindlin type. Inr. J. Numer. Mesh. Engng 26, 1169-1184 (1988).