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Vibration analysis of flat shells by using B spline functions
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VIBRATION ANALYSIS OF FLAT SHELLS BY USING B SPLINE FUNCTIONS SHEN PENG-CHENG~and
WAN
JIAN-GUO$
Hefei Polytechnic University, Hefei, Anhui, China (Received 24 January 1986)
Abstract-An efficient method is briefly described for vibration analysis of flat shells using spline functions. The cubic B, spline, quintic B, spline function and series which satisfy the boundary conditions are used to obtain the approximate solution of flat shells. Unified computational schemes suited for various types of boundary conditions are formulated. In comparision with the conventional finite element method and the finite strip method, the main features of the present method are higher accuracy, lesser unknowns, easy programming and economy in computer effort. The numerical examples are solved and compared with other methods.
INTRODUCTlON
agreement compared with the exact results and other numerical results.
A considerable amount of research on the vibration analysis of plates and shells has been carried out using various numerical methods, such as the finite FORMULATION USING SPLINE FUNGI-ION difference method, finite element method and the finite strip method. The displacement functions of flat shells are However, it has been found that the conventional defined in the form of the product of B,, B, spline finite element analysis may be unnecessary for the functions and beam functions as follows: plate and shell type structures having regular geometries. For such a kind of structure, some higher N+I r efficient numerical methods have been established, 4A,(~)~m(y)sinW + ~1 u=c c n--l m-l such as the finite strip method [2], which can reduce the size of problems. A combination of polynomial and harmonic functions which satisfy the boundary conditions are used in the finite strip analysis. It has NtI I the advantage of greatly reducing the number of 0 = C C hd#b(x>Ym(u)sin(~~ + e) n--, m-I system equations to be solved. This gives very economical solutions which are ideally suited to microcomputers. In recent years, some contributions to the solution Nt2 r procedure have been made by using B spline funcw = C C cdk(x)Y,(yW(wt +e) n--2 m-l tions for plate type structure. Mizusawa et al. [l, 31 and Chung-Tze [4] presented the procedures of the =Ld(x)J~LY(y)J(C)sin(ot+e), (1) Rayleigh-Ritz discrete approach and finite element method by using B spline functions for th’e vibration, where w is the frequency, e is the phase, and u, v and stability and static problems based on the variational w are displacement components. principle. The work published on vibration analysis of flat shells by using spline function method is not so extensive. The present paper deals with the determination of is the row of B, spline basis as shown in Appendix I. natural frequencies of flat shells using Bj, B5 spline LIL(x>J= W&)$-,(x). . . functions and beam functions which satisfy the boundary conditions; the spline functions can easily . . . ~N(x)~N+I(x)IL,~+~(x)I accommodate any type of boundary conditions of flat shells. In this paper, some illustrative examples solved is the row of B, spline basis shown as in Appendix I. here by microcomputer program demonstrate good LVY)J = V,(Y)Y*(Y) * *. P,(Y)1 t Associate Professor. $ Graduate Student. LY(Y>J = W,(Y)Y,(Y). . . y,(v)1
2
SHENPENG-CHENGand WAS JIAS-GLO
is the row of basic consistent with beam basic function which satisfies the boundary conditions of a flat shell. For instance, if two opposite edges are simply supported, we have,
The strain-displacement are defined as follows:
relations of the flat shells
(8)
(6) = [.il:f;, where the strain vector is
Y(0) = Y(L) = 0
it} = [c, ~:~I:%1%*%J.
Y”(0) = Y”(L) = 0
The differential
ym
matrix operator r
\“/
2
&
(/L(,=n,27r...mn).
(2)
[K]=
Y(0) = Y(L) = 0 Y’(0) = Y’(L) = 0 Y,,,=sin(y)-sin,($)-,,
c
0
If two opposite edges are fixed, we have
x [cos(y)
0
-
- coshr$)] The stress-strain
a
;_ cy
TV r
z
0
0
0
0
0
0
2m + 1
of flat shells is
1 R, 1 K 0
(10)
-- a: dx 2 a? -7 ay3: -a.+
relation of a flat shell is given by
5( -_ sin p,,, - sinh p, m cos p, - cash p,,, &=~~
(9)
(11)
(m=1,2...).
(3)
where the internal force vector {0) = [N,N,iV,zM,M*MJ.
For other cases of boundary conditions, the beam basic functions can be referred to [2]. L J represents a row matrix and @ is Kronecker product of matrices, P IO PI = k4,W.
(12)
The flexural rigidity matrix of flat shell is described by: (13)
{A} is column vector of spline node parameter.
(14) and the other two spline node parameter vectors {B} and {C} are similar in form to {A}. These parameters are all unknowns. Writing eqn (1) in matrix form, {f} = [s]{F}sin(wt + e).
From eqns (4), (7) and (8), {c} = [A][f]{F) sin(ot + e)
(4)
= [B]{F) sin(ot + e),
(15)
where {f } = [ucw]r
(5)
{F} = [{AIJ{WICIW
(6)
L~(~)JOL~(Y>_l bl = i
0
0
0
L~(x)J@LYY(Y)J
0
0
0
LW)JOLVY)J
(7)
Vibration analysis of flat shells using B spline functions
where r
L@‘(x)JOLf’b)J
LQ(~)J@LY’(Y)J
0
PI =
The functional
=P
=
n 2w
LQ(x)J@LI”(Y)J
$LWJOLWJ r
Ld’C~)J@LW)J
0
0
0 0
0
(16)
0
-L$“(x)JOLY(Y)J -L$(x)JOLY”(Y)J -2L$‘(x)JOLf”(v)J_
0
of the flat shell for vibration is In (24)
[JJ
W’Lcp JVlbp 1
-
$LWJBLWJ x
0
po*~(~*+u*+~*)dxdy
/.I
1,
(17)
where
JJ JJ [K,,]= Kpl =
WBpl%Wpl dx dl
PGI =
W~1T[4P~1dx dy
D'D,@F, + D'(l,')F,@Dy
0 = circular frequency
[Kw]=D'pE~@E,+D"12-p)E,@E~ p = material mass density
(25)
(18) (19)
= [B,]{F}sin(wr {&} =
+ e)
[K,J=
_5&$_2!&]’ [
= [B,]{F}sin(ot
+H,@J$+2(1-p)1,@4,].
+ e).
Substituting eqns (l), (20) and (21) into eqn (17), and using Hamiltons principle, we yield the following equation:
([Kl - ~‘PfI){F) = IQ.
The
The maSS matrix
IW =
(22)
P+I%I k dy
= Fx@F, &@F* 0
frequency equation of a flat shell is
IV4 - ~2bflI = 0,
(23)
W)
0 H,@H,
where
where 0, =
r L Ld’@)J24’(x)J
h
1 (27) ’
SHES PEN-CHENC and WAS JIAN-GUO
4
0
[~nlml= L Lx
=
j0
(32) 0
Lll/(xWLd’(-y)J dx For the flat cylindrical
H.x= L LWU’LIL(x)J dx s0
shells, we have
F&+&$D!$
[KC”]=
KY = I,
=
L
LJl’(~)J?_l(l’(x)J dx
s 0
J, =
50
LLWULVYx)J
(28)
dx.
The above matrices are shown in detailed form in Appendix II. If the two opposite edges are simply supported, the beam function is a group of harmonic functions, i.e. P,(v)
= cos W
Y,(y) = sin K,y Km = 7
(m = 1,2,. . . r),
with their properties of orthogonality, it results in matrices which have no coupling between the different terms and therefore a term by term analysis involving only small matrices can be performed and great savings in computer efforts can be achieved. Thus, the matrices [K] and [M] of the problem can be reduced to the following simple form:
(33)
For the flat doubIe curved shells, we have [K”]=D’F,K~~+D”‘~-p’D~~
t&,1 Ll
0
(K”,] = -DfpEyKm;+
D’(12-p)E:K,;
(2%
(30)
WI =
Wmml [Kzw.]=
Km1 =
(
5
+ s
.x Y
x Hy;+D
-Ku1 K.l Kl
[C’wl
_ symmetric
[Kfw]I
[KLI
(31) -
+ $
Y) L
H,K:,;-IJ:K+ L
FJ,K: 7 + 2(1 - p,I,Kk,
L
1.
(34)
Vibration analysis of fiat sheik using B spike functions Table 1. 01
0.
a3 (rad/sec)
Present method (spline function method) N=7 M= 1,2,3
0.28285 06
0.30285 07
0.5055 1 %
0.52489 @9
010
0.73930
0.75758
0.8244 I
0.969930
1.0563
Curve strip 121 Flat strip f21
0.284
0.301
0.509
0.527
0.572
0.285
0.305
0,512
0.530
0.573
Method
W1
% 0.57185
Table 2. 01
W2
@3 (radjsec)
% 0.69040
Method Present method
0.52835
(spline function method) N=7 M= 1,2,3 Reference f5lt
0.59151
0.59253
%
01
%
0.77307
0.90372
0.9073-t
q 0.77~70
09
WI0
1.10283
1.15263
0.5358471
+The lowest frequency was calculated from using the formulas (41), (40) and (39) of [5]. Table 3. 01
02
w3
04
WS
06
07
(radjsec)
Method
Present method 19.740461 49.343355 49.347221 64.383505 78.953353 91.05196 98.690213 (spline function method) N = 7, M = 1, 2, 3 Lower order strip 19.74 49.32 49.34 78.91 98.69 [2, p. 1341 Higher order strip 19.74 49.35 49.36 78.97 98.94 12,P. 1341 Exact solution [a]
19.74
49.35
49.35
NUMERICAL EXAMPLESt
According to the formulation given in the above section, a FORTRAN 77 microcomputer program has been developed to compute the frequencies of the flat shells. The program consists of a main program and some subprograms. It can be suited to compute the free vibration frequencies of the shallow shells, cylindricaf shells and plates which are simply supported on the two opposite edges and where the other edges are arbitrarily supported. Some illustrated examples of flat shells are solved by microcomputer program. The results are in good agreement with other methods 12, S] and comparisons are shown in Tables 1, 2 and 3. The calculation precision is dependent upon the numbers of meshes and terms of beam functions. ?Note: all E numbers refer to numbers fo~o~ng the decimal point, e.g. 28285268 + 00 = 0.2828526E f 00 etc.
-
78.95
Case I-Simply
-
98.69
supported flat cylindrical she&
N=7
M=l 2828526E + 9699300E + 2135019E+ 3934233E + 6165605E + 1~12~E + 1360623E + 1694344E + 2071059E + 5087199E +
00 00 01 01 01 02 02 02 02 02
M-2 505514lE + 00 1144712E+Ol 3462027E + 01 4364424E + 01 7368002E + 01 1060768E + 02 1474807E + 02 174324OE+ 02 2149853E + 02 51~~5E~O2
3028513E + 00 1489091E+Ol 2774413E + 01 4364379E + 01 7379229E + 01 1042822E + 02 1630274E + 02 I713837E + 02 2326987E + 02
5718532E f 00 1948333E + 01 3296417E+Ol 46712238 + 01 8048089E -I-01 1211520E+02 1632187E + 02 173254lE + 02 5008490E + 02
5248939E + 16605028 + 3896664E + 5526384E i8735403E + 1188286E + 164266OE+ 1781906E + 2385402E i-
7575750E + 00 2305042E f 01 4101147E f 01 7032978E -I-01 9321718E + 01 1276514E + 02 16738008 + 02 1785302E + 02 5023377E + 02
00 01 01 01 01 02 02 02 02
SHENPEW-CHENGand WANJIAS-GUO
6 M = 3 7393OOOE+ 00 143726lE + 01 3739132E + 01 6164285E -t- 01 9768933E + 01 118775lE+02 I647891E -t 02 1762928E + 02 2273340E i 02 5123840E -i- 02 Programmed STOP
8244164E + 1947886E + 4380331E + 7031329E + 1041627E + 1370176E + 1661936E f 1895894E + 2480864s +
00 01 01 01 02 02 02 02 02
1058419E + 01 25893148 + 01 5845OOOE+ 01 82744OOE-I-0 I 1151465E+02 1397730E + 02 1741609E + 02 1927429s c 02 504822lE + 02
t
=o.olgI
E=f u=o.3 i=r
Fig. 2.
Case ~i~-Sirnp~~
supported
plate
T
Fig. 1.
4
1 Case II-Simply
N=7 M=l 5283480E c 00 1102836E+Ol 2204107s -t 01 39820378 + 01 6162656E + 01 1001120E+02 I361467E t 02 1694981E + 02 2071634E + 02 5087354E + 02 A! = 2 5915099E + 00 1258538s + 01 34916338 -i- 01 43567198 + 01 7398870E + 01 1060515E-tO2 1475729E + 02 1739624E c 02 2150542E tO2 5100798E i 02 M = 3 7706985E f 00 1524319E +01 3754712E + 01 6161190E+Ol 97637278 + 01 1189566E + 02 1~8738E + 02 1760925E + 02 2274166E -t- 02 5123218E + 02 Programmed STOP
supported
double
curved fiat
she/i
59252578 f 00 1580393E -I-01 2755280E + 01 4357076E + 01 7398827E + 01 1044017E + 02 1627822E i- 02 1711636E-t.02 2327461E -i 02
77306958 + 1948333E + 33376688 f 47126198 f 80466558 + 1211493E + 1635587E + 17357978 + 5008647E i-
00 01 01 01 01 02 02 02 02
6904006E f 00 1744066E +Ol 3896665E + 01 S5f 1105E + 01 87314968 i 01 1I8%35E + 02 16419938 + 02 1785991E + 02 2385991E + 02
9074485E i2367085E + 4130892s + 70267408 + 9342391E + 1276408E + 1675422E -t 17868038 + 50235328 +
00 01 01 01 01 02 02 02 02
9037152E + 20194548 + 4387887E + 7025022E + 10438898 + 1369970E c 16616308 + 18995378 e 2481614‘S+
11526298 + 01 26427568 + 01 58450068 I- 01 8268006E -t 01 115106oE+02 1399038E + 02 17428768 + 02 1928174s -t 02 5048373s + 02
00 01 01 01 02 02 02 02 02
L= I t =0-J
E=f If=o.j
Fig. 3.
CONCLUSION
The cubic B, and quintic BS spline functions and beam functions are used to construct the displacement components of flat shells. Because the functions have good properties with piecewise ~lynomial and orthogonality, the present method is of lesser unknowns and higher accuracy and the calculations for the practical engineering problems of flat shells having regular geometries can be performed on a microcomputer.
REFERENCES
1. T. Mizusatia, T. Kajita and M. Naruoka, Vibration of skew plates by using B-spline functions. J. Sound Yibr. 301-308 (1979). 2. Y. K. Cheung, Finite Strip Metlzod in Structural Anatysis. Pergamon press, Oxford (1976). 3. T. Mizusawa, T. Kajita and M. Narucka, Vibration of stiffened skew plates by using B-spline functions. Comput. Struct. 10, 821-826 (1979). 4. S. Chung Tze, On spline finite element method. Mathematical Numeuicalsit&a 1 (1979). 5. D. Bucco and J. Maxumdar, Estimation of the fundamental frequencies of shallow shell. Compuf. Srrucr. 17, 441-447 (1983). 6. G. B. Warburton, The vibration of rectangular plates. Proc. Inst. Mech. Engrs 371-385 (1954).
Vibration analysis of flat shells using B spline functions APPENDIX I The expressions of cubic spline are: XE[-2,--I] XE[-I.01 x Ek-4II xEiL21 ix ;>2.
I” ;;;: q,(x) = l/6
(z-x)’ (2 - x)’ 0
The expressions of quintic spline are:
-6(x + 2)r -6(x+2)’
x a[-3, -2) .re(-2, -I] xe[-I.01 x l LO,11
+15(x + 1)’
cp,(x) = l/120
XE[1,21 x E P, 31 1.r/)3.
.O
The expression of cubic B-spline function for N equal divisions in the interval [JC~, xN] is
4x1 = 1 C,Q,[(x -x)/h - i), ,--I
where i= -I,O, I,2 ,..., N,N+l,h=(x,“-x,j/N x,.x,, x2, . . . ,x,,, are spline nodes
c, is the parameter of spline nodes. For the simplicity of the treatment of boundary conditions, the above equation can be written as follows: N+I
1
$tx)=
ci4i(xX
i--l
where, d,(x) is the basic function of a cubic B-spline W)
= &lh
- 9;
when N 2 4, 4-O)
= &x/h
+ 1)
h,(x) = cp,(xlh) - 4rp,(xlh + 1) 4,(x) = cp,(x/h- 1)- l/%(x/h)
+ &x/h + 1)
42(x) = cp,(xlh - 2)
‘#‘rv-#)=Q,(X/h-N+2) +N_ ,(x) = 9,(x/h - N + I) - 1/2cp,(x/h - N) + cp,(x/h - N - 1) &dX) 4N+ The
I(X)
= rp,(xlh - N) -%(x/h =
&lh
-N
-
- N - 1)
1).
expression of quintic B, spline function for N equal divisions in the interval [xr,,xN] is N+3 4x)
=,
r* _
Cillhh
when N > 6, +_r(.r) = cp,(xlh + 2) +-Lx) = &x/h
+ 1) -26%(x/h
Il&) = W~Q,(X/~
+ 2)
+ 2) - 33/8%(x/h
G,(x) = $+x/h
+ 1) -26/33%(x/h)
&(x)
+ 2) - 1/33%(x/h)
= &x/h
G,(x) = rp,(x/h - 3)
+ 1) + %(x/h)
+ %(x/h + %(x/h
- 1) - 2)
SHENPENG-CHENGand W.w JIAN-GCO Ijt,v.&) = &x/h
-h’ f 3)
+N_ *(x) = c,@xjh-A’ c 2) - 1/33q$c/h - N) f 9,(x/h - N + 2) $N_ ,(x) = cp,(xjh - N - 1) -26/339,(x/h
- N) +9,(x/h
- N -t-
1)
$#(x) = 165/49,(x/h - N - 2) - 331’8q+(x/h- N - 1) + 9,(x/h - N) +,Y+,(x) = q+/h 3/t,,.h)
- N - I) - 26~~(~~h - iti - 2)
= cp,(.ylh - N - 2).
APPENDIX II l/252 Fx = h
71720 31/315 sym
31/1~0 S/48 183/560
1~5~0 2911260 28311260 151/31!.
l/5040 239,‘10080 397/168!
i l/20 Dx = l/h
- 1?/120 213 sym
-1112
EX=
311720 0
antisym
252 2561 94144
- 191240 1140 27140
l/720 13/180
l/720
0
0 .. 1~13~
49/l? 37/144
1322.87875 229765.8185 2508619.27 746326.6035
sym
1190 .. l/720
374.30 89010.45 1573369.69 8~7284.296 15599958.86
1/72p I
1 2010 I 144279.75 2031.875 2085264.5 151033.88 9733489.6 2203426.3 157724248 9738114 ‘. ‘. 0 0 0 1 2036 152637 ‘.
70
- 769 13376
Ix = l~(362880~) vm
-987.875 4395 48296.25
-641.42 -8744.72 30534.90 109641.34
l/5?0 .I
- If120 _ 471240 --l/8 ..
51288 1331180
2361.875 241941 1460247.75
Hx = hl3~!68~
-l/120 -l/6 -l/30 213 ‘.
I;5040 1141 ’
- 193.93 -6811.18 -20683.15 40410.185 139081.675
-1 - 526 - 11583.75 -49300.90 6081.27 1359!_r 0 0 0 0 0 0” -1 -500 - 13605 ..
0 0 0 0 1 2036 ...
i 2035.21 152636.96 2203488 ‘,
0 -I 0 0 0 0 I . ..
-1 -495.875 - 13212.06 - 59504.84 5760.. 0 0 0 0
’
- 1 -499.21 - 13604.96 - 59520
Vibration analysis of flat shells using B spline functions
‘56 769
- 1247 -13376
987.875 -439s -48296.25
641.42 8744.72 - 30534.90 - 109641.34
J.r = i~(36288Ok) sYm
193.93 6811.18 20683.1415 -40410.185 - 139081.675
1.
.9
I I
526 11583.75 49300.90 -6081.27 - 135912 ..
495875 I 13212.06 #?.:!I 59504.84 13604.96 - 567: 59510 .. 0
:: 0
0 0 01
13605
so0
0 0 0:: ~ I
:oo
‘.
‘.
1
Kx = 1,‘(5040k’)
9664
-7659 11967.75
- 5579.63 3368.36 8038.25895
sym 20
-431
Gx = i/362880
279.875
239.24
‘56 769 987.875 641.4242 193.9394
1
97 5080 19314 24138.91 18970.7212 498 1
- 1345.90 - 1663.034 313.716253 5425.30027 83.39
90 279.75 -2080.90 - 1826.72 5208 1
21s 3417 25816.375 87928 83896 14358 501.5
1
..
Lx = 1140320
-21.0 -351.0 -564.125 -409.79 - 131.0303 -1
71.0 536.0 -2128.0 -5147.3939 -3280.5151 - 242 -1
18.5 1081.0 3261.375 -2303.8182 -9782.95 - 3924 -245.5
1 111.875 626.60 -1635.5 - 1906
1 115.21 716.96 -16!2
1 116 7!7
-.
..
.
1 476 12578.5 77226.6661 155748.3333 88234 14608 so2 ..
I 220 3072.5 8384.2424 - 121.6061 -11326 -4046 -246
1 497.875 14213.4848 88218.7878 156190 88234 14608
1 116 ._.
1 501.2121 14607.97 88234 156190 88234
t so2 14608 88231 156190
0
0
0
0
0
0
1 so2 14608 88234 ..
0 1 SO2 14608 ..
0 0 0 I 502
0 0 0 0 0 0 I
:41.*75 3853.1818 11318.5454 0 -11326 -4046 ‘.
1 246 4046 II?26
.J
..
..
0 0 0 0
I ._ :
z
1 245.2121 4045.9697 11326 0 -11326 ..
0
1 246 4046 I1326 0 ..
0
0 0
0 0
8
0 0
0 0
0 1 246 41?46 .*
0 0 1 246 ‘.
0. 1 0 0 ! ..
Note: For the matrices Dx and Fx, the last three _WWSand the last three columns symmetrize the first three rows and the first three columns. From the fourth row from the top to the fourth from bottom row, the form of every row is the same. For the matrix Ex, the last three rows and the last three columns antisymmetrize the first three rows and the first three columns. From the fourth row from the top to the fourth from bottom row, the form of every row is the same. For the matrices Hx, Ix, Jx and Kx, the last five rows and the last five columns symmetrize the first five rows and the first five columns. From the sixth row from the top to the sixth row from the bottom, the form of every row is the same. For
IO
SHES PWG-CHENGand WAS
JIAN-GUO
the matrix Gx. the last five rows and the last three columns symmetrize the first five rows and the first three columns. From the eighth row to the sixth from bottom row, the form of every row is the same. For the matrix Lx, the last five rows and the last three columns antisymmetrize the first five rows and the first three columns. From the eighth row to the sixth from bottom row, the form of every row is the same.
VIBRATION ANALYSIS OF FLAT SHELLS BY USING B SPLINE FUNCTIONS SHEN PENG-CHENG~and
WAN
JIAN-GUO$
Hefei Polytechnic University, Hefei, Anhui, China (Received 24 January 1986)
Abstract-An efficient method is briefly described for vibration analysis of flat shells using spline functions. The cubic B, spline, quintic B, spline function and series which satisfy the boundary conditions are used to obtain the approximate solution of flat shells. Unified computational schemes suited for various types of boundary conditions are formulated. In comparision with the conventional finite element method and the finite strip method, the main features of the present method are higher accuracy, lesser unknowns, easy programming and economy in computer effort. The numerical examples are solved and compared with other methods.
INTRODUCTlON
agreement compared with the exact results and other numerical results.
A considerable amount of research on the vibration analysis of plates and shells has been carried out using various numerical methods, such as the finite FORMULATION USING SPLINE FUNGI-ION difference method, finite element method and the finite strip method. The displacement functions of flat shells are However, it has been found that the conventional defined in the form of the product of B,, B, spline finite element analysis may be unnecessary for the functions and beam functions as follows: plate and shell type structures having regular geometries. For such a kind of structure, some higher N+I r efficient numerical methods have been established, 4A,(~)~m(y)sinW + ~1 u=c c n--l m-l such as the finite strip method [2], which can reduce the size of problems. A combination of polynomial and harmonic functions which satisfy the boundary conditions are used in the finite strip analysis. It has NtI I the advantage of greatly reducing the number of 0 = C C hd#b(x>Ym(u)sin(~~ + e) n--, m-I system equations to be solved. This gives very economical solutions which are ideally suited to microcomputers. In recent years, some contributions to the solution Nt2 r procedure have been made by using B spline funcw = C C cdk(x)Y,(yW(wt +e) n--2 m-l tions for plate type structure. Mizusawa et al. [l, 31 and Chung-Tze [4] presented the procedures of the =Ld(x)J~LY(y)J(C)sin(ot+e), (1) Rayleigh-Ritz discrete approach and finite element method by using B spline functions for th’e vibration, where w is the frequency, e is the phase, and u, v and stability and static problems based on the variational w are displacement components. principle. The work published on vibration analysis of flat shells by using spline function method is not so extensive. The present paper deals with the determination of is the row of B, spline basis as shown in Appendix I. natural frequencies of flat shells using Bj, B5 spline LIL(x>J= W&)$-,(x). . . functions and beam functions which satisfy the boundary conditions; the spline functions can easily . . . ~N(x)~N+I(x)IL,~+~(x)I accommodate any type of boundary conditions of flat shells. In this paper, some illustrative examples solved is the row of B, spline basis shown as in Appendix I. here by microcomputer program demonstrate good LVY)J = V,(Y)Y*(Y) * *. P,(Y)1 t Associate Professor. $ Graduate Student. LY(Y>J = W,(Y)Y,(Y). . . y,(v)1
2
SHENPENG-CHENGand WAS JIAS-GLO
is the row of basic consistent with beam basic function which satisfies the boundary conditions of a flat shell. For instance, if two opposite edges are simply supported, we have,
The strain-displacement are defined as follows:
relations of the flat shells
(8)
(6) = [.il:f;, where the strain vector is
Y(0) = Y(L) = 0
it} = [c, ~:~I:%1%*%J.
Y”(0) = Y”(L) = 0
The differential
ym
matrix operator r
\“/
2
&
(/L(,=n,27r...mn).
(2)
[K]=
Y(0) = Y(L) = 0 Y’(0) = Y’(L) = 0 Y,,,=sin(y)-sin,($)-,,
c
0
If two opposite edges are fixed, we have
x [cos(y)
0
-
- coshr$)] The stress-strain
a
;_ cy
TV r
z
0
0
0
0
0
0
2m + 1
of flat shells is
1 R, 1 K 0
(10)
-- a: dx 2 a? -7 ay3: -a.+
relation of a flat shell is given by
5( -_ sin p,,, - sinh p, m cos p, - cash p,,, &=~~
(9)
(11)
(m=1,2...).
(3)
where the internal force vector {0) = [N,N,iV,zM,M*MJ.
For other cases of boundary conditions, the beam basic functions can be referred to [2]. L J represents a row matrix and @ is Kronecker product of matrices, P IO PI = k4,W.
(12)
The flexural rigidity matrix of flat shell is described by: (13)
{A} is column vector of spline node parameter.
(14) and the other two spline node parameter vectors {B} and {C} are similar in form to {A}. These parameters are all unknowns. Writing eqn (1) in matrix form, {f} = [s]{F}sin(wt + e).
From eqns (4), (7) and (8), {c} = [A][f]{F) sin(ot + e)
(4)
= [B]{F) sin(ot + e),
(15)
where {f } = [ucw]r
(5)
{F} = [{AIJ{WICIW
(6)
L~(~)JOL~(Y>_l bl = i
0
0
0
L~(x)J@LYY(Y)J
0
0
0
LW)JOLVY)J
(7)
Vibration analysis of flat shells using B spline functions
where r
L@‘(x)JOLf’b)J
LQ(~)J@LY’(Y)J
0
PI =
The functional
=P
=
n 2w
LQ(x)J@LI”(Y)J
$LWJOLWJ r
Ld’C~)J@LW)J
0
0
0 0
0
(16)
0
-L$“(x)JOLY(Y)J -L$(x)JOLY”(Y)J -2L$‘(x)JOLf”(v)J_
0
of the flat shell for vibration is In (24)
[JJ
W’Lcp JVlbp 1
-
$LWJBLWJ x
0
po*~(~*+u*+~*)dxdy
/.I
1,
(17)
where
JJ JJ [K,,]= Kpl =
WBpl%Wpl dx dl
PGI =
W~1T[4P~1dx dy
D'D,@F, + D'(l,')F,@Dy
0 = circular frequency
[Kw]=D'pE~@E,+D"12-p)E,@E~ p = material mass density
(25)
(18) (19)
= [B,]{F}sin(wr {&} =
+ e)
[K,J=
_5&$_2!&]’ [
= [B,]{F}sin(ot
+H,@J$+2(1-p)1,@4,].
+ e).
Substituting eqns (l), (20) and (21) into eqn (17), and using Hamiltons principle, we yield the following equation:
([Kl - ~‘PfI){F) = IQ.
The
The maSS matrix
IW =
(22)
P+I%I k dy
= Fx@F, &@F* 0
frequency equation of a flat shell is
IV4 - ~2bflI = 0,
(23)
W)
0 H,@H,
where
where 0, =
r L Ld’@)J24’(x)J
h
1 (27) ’
SHES PEN-CHENC and WAS JIAN-GUO
4
0
[~nlml= L Lx
=
j0
(32) 0
Lll/(xWLd’(-y)J dx For the flat cylindrical
H.x= L LWU’LIL(x)J dx s0
shells, we have
F&+&$D!$
[KC”]=
KY = I,
=
L
LJl’(~)J?_l(l’(x)J dx
s 0
J, =
50
LLWULVYx)J
(28)
dx.
The above matrices are shown in detailed form in Appendix II. If the two opposite edges are simply supported, the beam function is a group of harmonic functions, i.e. P,(v)
= cos W
Y,(y) = sin K,y Km = 7
(m = 1,2,. . . r),
with their properties of orthogonality, it results in matrices which have no coupling between the different terms and therefore a term by term analysis involving only small matrices can be performed and great savings in computer efforts can be achieved. Thus, the matrices [K] and [M] of the problem can be reduced to the following simple form:
(33)
For the flat doubIe curved shells, we have [K”]=D’F,K~~+D”‘~-p’D~~
t&,1 Ll
0
(K”,] = -DfpEyKm;+
D’(12-p)E:K,;
(2%
(30)
WI =
Wmml [Kzw.]=
Km1 =
(
5
+ s
.x Y
x Hy;+D
-Ku1 K.l Kl
[C’wl
_ symmetric
[Kfw]I
[KLI
(31) -
+ $
Y) L
H,K:,;-IJ:K+ L
FJ,K: 7 + 2(1 - p,I,Kk,
L
1.
(34)
Vibration analysis of fiat sheik using B spike functions Table 1. 01
0.
a3 (rad/sec)
Present method (spline function method) N=7 M= 1,2,3
0.28285 06
0.30285 07
0.5055 1 %
0.52489 @9
010
0.73930
0.75758
0.8244 I
0.969930
1.0563
Curve strip 121 Flat strip f21
0.284
0.301
0.509
0.527
0.572
0.285
0.305
0,512
0.530
0.573
Method
W1
% 0.57185
Table 2. 01
W2
@3 (radjsec)
% 0.69040
Method Present method
0.52835
(spline function method) N=7 M= 1,2,3 Reference f5lt
0.59151
0.59253
%
01
%
0.77307
0.90372
0.9073-t
q 0.77~70
09
WI0
1.10283
1.15263
0.5358471
+The lowest frequency was calculated from using the formulas (41), (40) and (39) of [5]. Table 3. 01
02
w3
04
WS
06
07
(radjsec)
Method
Present method 19.740461 49.343355 49.347221 64.383505 78.953353 91.05196 98.690213 (spline function method) N = 7, M = 1, 2, 3 Lower order strip 19.74 49.32 49.34 78.91 98.69 [2, p. 1341 Higher order strip 19.74 49.35 49.36 78.97 98.94 12,P. 1341 Exact solution [a]
19.74
49.35
49.35
NUMERICAL EXAMPLESt
According to the formulation given in the above section, a FORTRAN 77 microcomputer program has been developed to compute the frequencies of the flat shells. The program consists of a main program and some subprograms. It can be suited to compute the free vibration frequencies of the shallow shells, cylindricaf shells and plates which are simply supported on the two opposite edges and where the other edges are arbitrarily supported. Some illustrated examples of flat shells are solved by microcomputer program. The results are in good agreement with other methods 12, S] and comparisons are shown in Tables 1, 2 and 3. The calculation precision is dependent upon the numbers of meshes and terms of beam functions. ?Note: all E numbers refer to numbers fo~o~ng the decimal point, e.g. 28285268 + 00 = 0.2828526E f 00 etc.
-
78.95
Case I-Simply
-
98.69
supported flat cylindrical she&
N=7
M=l 2828526E + 9699300E + 2135019E+ 3934233E + 6165605E + 1~12~E + 1360623E + 1694344E + 2071059E + 5087199E +
00 00 01 01 01 02 02 02 02 02
M-2 505514lE + 00 1144712E+Ol 3462027E + 01 4364424E + 01 7368002E + 01 1060768E + 02 1474807E + 02 174324OE+ 02 2149853E + 02 51~~5E~O2
3028513E + 00 1489091E+Ol 2774413E + 01 4364379E + 01 7379229E + 01 1042822E + 02 1630274E + 02 I713837E + 02 2326987E + 02
5718532E f 00 1948333E + 01 3296417E+Ol 46712238 + 01 8048089E -I-01 1211520E+02 1632187E + 02 173254lE + 02 5008490E + 02
5248939E + 16605028 + 3896664E + 5526384E i8735403E + 1188286E + 164266OE+ 1781906E + 2385402E i-
7575750E + 00 2305042E f 01 4101147E f 01 7032978E -I-01 9321718E + 01 1276514E + 02 16738008 + 02 1785302E + 02 5023377E + 02
00 01 01 01 01 02 02 02 02
SHENPEW-CHENGand WANJIAS-GUO
6 M = 3 7393OOOE+ 00 143726lE + 01 3739132E + 01 6164285E -t- 01 9768933E + 01 118775lE+02 I647891E -t 02 1762928E + 02 2273340E i 02 5123840E -i- 02 Programmed STOP
8244164E + 1947886E + 4380331E + 7031329E + 1041627E + 1370176E + 1661936E f 1895894E + 2480864s +
00 01 01 01 02 02 02 02 02
1058419E + 01 25893148 + 01 5845OOOE+ 01 82744OOE-I-0 I 1151465E+02 1397730E + 02 1741609E + 02 1927429s c 02 504822lE + 02
t
=o.olgI
E=f u=o.3 i=r
Fig. 2.
Case ~i~-Sirnp~~
supported
plate
T
Fig. 1.
4
1 Case II-Simply
N=7 M=l 5283480E c 00 1102836E+Ol 2204107s -t 01 39820378 + 01 6162656E + 01 1001120E+02 I361467E t 02 1694981E + 02 2071634E + 02 5087354E + 02 A! = 2 5915099E + 00 1258538s + 01 34916338 -i- 01 43567198 + 01 7398870E + 01 1060515E-tO2 1475729E + 02 1739624E c 02 2150542E tO2 5100798E i 02 M = 3 7706985E f 00 1524319E +01 3754712E + 01 6161190E+Ol 97637278 + 01 1189566E + 02 1~8738E + 02 1760925E + 02 2274166E -t- 02 5123218E + 02 Programmed STOP
supported
double
curved fiat
she/i
59252578 f 00 1580393E -I-01 2755280E + 01 4357076E + 01 7398827E + 01 1044017E + 02 1627822E i- 02 1711636E-t.02 2327461E -i 02
77306958 + 1948333E + 33376688 f 47126198 f 80466558 + 1211493E + 1635587E + 17357978 + 5008647E i-
00 01 01 01 01 02 02 02 02
6904006E f 00 1744066E +Ol 3896665E + 01 S5f 1105E + 01 87314968 i 01 1I8%35E + 02 16419938 + 02 1785991E + 02 2385991E + 02
9074485E i2367085E + 4130892s + 70267408 + 9342391E + 1276408E + 1675422E -t 17868038 + 50235328 +
00 01 01 01 01 02 02 02 02
9037152E + 20194548 + 4387887E + 7025022E + 10438898 + 1369970E c 16616308 + 18995378 e 2481614‘S+
11526298 + 01 26427568 + 01 58450068 I- 01 8268006E -t 01 115106oE+02 1399038E + 02 17428768 + 02 1928174s -t 02 5048373s + 02
00 01 01 01 02 02 02 02 02
L= I t =0-J
E=f If=o.j
Fig. 3.
CONCLUSION
The cubic B, and quintic BS spline functions and beam functions are used to construct the displacement components of flat shells. Because the functions have good properties with piecewise ~lynomial and orthogonality, the present method is of lesser unknowns and higher accuracy and the calculations for the practical engineering problems of flat shells having regular geometries can be performed on a microcomputer.
REFERENCES
1. T. Mizusatia, T. Kajita and M. Naruoka, Vibration of skew plates by using B-spline functions. J. Sound Yibr. 301-308 (1979). 2. Y. K. Cheung, Finite Strip Metlzod in Structural Anatysis. Pergamon press, Oxford (1976). 3. T. Mizusawa, T. Kajita and M. Narucka, Vibration of stiffened skew plates by using B-spline functions. Comput. Struct. 10, 821-826 (1979). 4. S. Chung Tze, On spline finite element method. Mathematical Numeuicalsit&a 1 (1979). 5. D. Bucco and J. Maxumdar, Estimation of the fundamental frequencies of shallow shell. Compuf. Srrucr. 17, 441-447 (1983). 6. G. B. Warburton, The vibration of rectangular plates. Proc. Inst. Mech. Engrs 371-385 (1954).
Vibration analysis of flat shells using B spline functions APPENDIX I The expressions of cubic spline are: XE[-2,--I] XE[-I.01 x Ek-4II xEiL21 ix ;>2.
I” ;;;: q,(x) = l/6
(z-x)’ (2 - x)’ 0
The expressions of quintic spline are:
-6(x + 2)r -6(x+2)’
x a[-3, -2) .re(-2, -I] xe[-I.01 x l LO,11
+15(x + 1)’
cp,(x) = l/120
XE[1,21 x E P, 31 1.r/)3.
.O
The expression of cubic B-spline function for N equal divisions in the interval [JC~, xN] is
4x1 = 1 C,Q,[(x -x)/h - i), ,--I
where i= -I,O, I,2 ,..., N,N+l,h=(x,“-x,j/N x,.x,, x2, . . . ,x,,, are spline nodes
c, is the parameter of spline nodes. For the simplicity of the treatment of boundary conditions, the above equation can be written as follows: N+I
1
$tx)=
ci4i(xX
i--l
where, d,(x) is the basic function of a cubic B-spline W)
= &lh
- 9;
when N 2 4, 4-O)
= &x/h
+ 1)
h,(x) = cp,(xlh) - 4rp,(xlh + 1) 4,(x) = cp,(x/h- 1)- l/%(x/h)
+ &x/h + 1)
42(x) = cp,(xlh - 2)
‘#‘rv-#)=Q,(X/h-N+2) +N_ ,(x) = 9,(x/h - N + I) - 1/2cp,(x/h - N) + cp,(x/h - N - 1) &dX) 4N+ The
I(X)
= rp,(xlh - N) -%(x/h =
&lh
-N
-
- N - 1)
1).
expression of quintic B, spline function for N equal divisions in the interval [xr,,xN] is N+3 4x)
=,
r* _
Cillhh
when N > 6, +_r(.r) = cp,(xlh + 2) +-Lx) = &x/h
+ 1) -26%(x/h
Il&) = W~Q,(X/~
+ 2)
+ 2) - 33/8%(x/h
G,(x) = $+x/h
+ 1) -26/33%(x/h)
&(x)
+ 2) - 1/33%(x/h)
= &x/h
G,(x) = rp,(x/h - 3)
+ 1) + %(x/h)
+ %(x/h + %(x/h
- 1) - 2)
SHENPENG-CHENGand W.w JIAN-GCO Ijt,v.&) = &x/h
-h’ f 3)
+N_ *(x) = c,@xjh-A’ c 2) - 1/33q$c/h - N) f 9,(x/h - N + 2) $N_ ,(x) = cp,(xjh - N - 1) -26/339,(x/h
- N) +9,(x/h
- N -t-
1)
$#(x) = 165/49,(x/h - N - 2) - 331’8q+(x/h- N - 1) + 9,(x/h - N) +,Y+,(x) = q+/h 3/t,,.h)
- N - I) - 26~~(~~h - iti - 2)
= cp,(.ylh - N - 2).
APPENDIX II l/252 Fx = h
71720 31/315 sym
31/1~0 S/48 183/560
1~5~0 2911260 28311260 151/31!.
l/5040 239,‘10080 397/168!
i l/20 Dx = l/h
- 1?/120 213 sym
-1112
EX=
311720 0
antisym
252 2561 94144
- 191240 1140 27140
l/720 13/180
l/720
0
0 .. 1~13~
49/l? 37/144
1322.87875 229765.8185 2508619.27 746326.6035
sym
1190 .. l/720
374.30 89010.45 1573369.69 8~7284.296 15599958.86
1/72p I
1 2010 I 144279.75 2031.875 2085264.5 151033.88 9733489.6 2203426.3 157724248 9738114 ‘. ‘. 0 0 0 1 2036 152637 ‘.
70
- 769 13376
Ix = l~(362880~) vm
-987.875 4395 48296.25
-641.42 -8744.72 30534.90 109641.34
l/5?0 .I
- If120 _ 471240 --l/8 ..
51288 1331180
2361.875 241941 1460247.75
Hx = hl3~!68~
-l/120 -l/6 -l/30 213 ‘.
I;5040 1141 ’
- 193.93 -6811.18 -20683.15 40410.185 139081.675
-1 - 526 - 11583.75 -49300.90 6081.27 1359!_r 0 0 0 0 0 0” -1 -500 - 13605 ..
0 0 0 0 1 2036 ...
i 2035.21 152636.96 2203488 ‘,
0 -I 0 0 0 0 I . ..
-1 -495.875 - 13212.06 - 59504.84 5760.. 0 0 0 0
’
- 1 -499.21 - 13604.96 - 59520
Vibration analysis of flat shells using B spline functions
‘56 769
- 1247 -13376
987.875 -439s -48296.25
641.42 8744.72 - 30534.90 - 109641.34
J.r = i~(36288Ok) sYm
193.93 6811.18 20683.1415 -40410.185 - 139081.675
1.
.9
I I
526 11583.75 49300.90 -6081.27 - 135912 ..
495875 I 13212.06 #?.:!I 59504.84 13604.96 - 567: 59510 .. 0
:: 0
0 0 01
13605
so0
0 0 0:: ~ I
:oo
‘.
‘.
1
Kx = 1,‘(5040k’)
9664
-7659 11967.75
- 5579.63 3368.36 8038.25895
sym 20
-431
Gx = i/362880
279.875
239.24
‘56 769 987.875 641.4242 193.9394
1
97 5080 19314 24138.91 18970.7212 498 1
- 1345.90 - 1663.034 313.716253 5425.30027 83.39
90 279.75 -2080.90 - 1826.72 5208 1
21s 3417 25816.375 87928 83896 14358 501.5
1
..
Lx = 1140320
-21.0 -351.0 -564.125 -409.79 - 131.0303 -1
71.0 536.0 -2128.0 -5147.3939 -3280.5151 - 242 -1
18.5 1081.0 3261.375 -2303.8182 -9782.95 - 3924 -245.5
1 111.875 626.60 -1635.5 - 1906
1 115.21 716.96 -16!2
1 116 7!7
-.
..
.
1 476 12578.5 77226.6661 155748.3333 88234 14608 so2 ..
I 220 3072.5 8384.2424 - 121.6061 -11326 -4046 -246
1 497.875 14213.4848 88218.7878 156190 88234 14608
1 116 ._.
1 501.2121 14607.97 88234 156190 88234
t so2 14608 88231 156190
0
0
0
0
0
0
1 so2 14608 88234 ..
0 1 SO2 14608 ..
0 0 0 I 502
0 0 0 0 0 0 I
:41.*75 3853.1818 11318.5454 0 -11326 -4046 ‘.
1 246 4046 II?26
.J
..
..
0 0 0 0
I ._ :
z
1 245.2121 4045.9697 11326 0 -11326 ..
0
1 246 4046 I1326 0 ..
0
0 0
0 0
8
0 0
0 0
0 1 246 41?46 .*
0 0 1 246 ‘.
0. 1 0 0 ! ..
Note: For the matrices Dx and Fx, the last three _WWSand the last three columns symmetrize the first three rows and the first three columns. From the fourth row from the top to the fourth from bottom row, the form of every row is the same. For the matrix Ex, the last three rows and the last three columns antisymmetrize the first three rows and the first three columns. From the fourth row from the top to the fourth from bottom row, the form of every row is the same. For the matrices Hx, Ix, Jx and Kx, the last five rows and the last five columns symmetrize the first five rows and the first five columns. From the sixth row from the top to the sixth row from the bottom, the form of every row is the same. For
IO
SHES PWG-CHENGand WAS
JIAN-GUO
the matrix Gx. the last five rows and the last three columns symmetrize the first five rows and the first three columns. From the eighth row to the sixth from bottom row, the form of every row is the same. For the matrix Lx, the last five rows and the last three columns antisymmetrize the first five rows and the first three columns. From the eighth row to the sixth from bottom row, the form of every row is the same.