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Free vibrations of shells of revolution using ring finite elements
Int. J . Meek. 8ci. Pergamon Press Ltd. 1967. Vol. 9, pp. 559-570. Printed in Great Britain
F R E E V I B R A T I O N S OF SHELLS OF R E V O L U T I O N USING R I N G F I N I T E ELEMENTS J. J. WEBSTER D e p a r t m e n t of Mechanical Engineering, University of Nottingham
(Received 9 February 1967) S u m m a r y - - I m p r o v e m e n t s in the ring finite element analysis, for shells of revolution, obtained b y extending the polynomials representing the displacements are illustrated b y comparing the natural frequencies of three shells obtained from this improved analysis with those obtained b y the usual method. The mass and stiffness matrices and their derivation for the elements with extended displacement functions are given.
h l n
{q} r rs~ r e 8
t U~ V, '/0 Un~ Vn~ Wn
{x} Z
[B] E
[K] R T
IT]
U,.,V,.,W,.
V Ot
0
P
~k 03
q~ k(i,j) [B-l] [B-~]~"
NOTATION axial distance from apex of element with closed end, see Fig. 2 meridional length of element number of cirenmferential waves column vector of generalized co-ordinates radius principal radii of curvature meridional co-ordinate shell thickness tangential, meridional and normal displacements amplitudes of the n t h harmonic components of the displacements defined in equation (1) column vector of displacement function coefficients normal co-ordinate, through shell thickness transformation m a t r i x relating {q} to {x} Young's modulus square m a t r i x in strain energy expression radius kinetic energy square m a t r i x in kinetic energy expression coefficients of ruth terms in displacement function strain energy angle, defined in Fig. 2 meridional, tangential and shear strains circumferential co-ordinate Poisson's ratio density angle between normal to shell surface and axis of shell circular frequency i t h element of column vector {q} j t h element of i t h row of m a t r i x [K] inverse of m a t r i x [B] transpose of inverse of [B] INTRODUCTION
FINITE element methods are used extensively to solve both static and dynamic problems for elastic structures.
For shells of revolution the structure has been 559
560
J . J . W~.BS~R
idealized either b y fiat plate elements or more usually by ring elements bounded b y surfaces of constant meridional co-ordinate. This paper is concerned with the latter approach. In the earlier work (Refs. 1, 2, 3) truncated cone elements were used, even for shells with meridional curvature. Large numbers of these elements were sometimes required to give a reasonable approximation to the shell geometry and recently 4,5 elements with meridional curvature have been developed. Exact expressions for the influence coefficients for some elements subjected to axisymmetric edge loads have been derived, 2 b u t in general the displacements of the middle surface of the elements are represented by a number of functions of the meridional co-ordinate. The coefficients of these functions, and in the free vibration problem the natural frequencies, are in effect determined by finding their values which make the relevant energy integral stationary. A typical displacement, say the normal displacement w, is represented by:
w= •
ofmo(~)+ [Wm,Jm~(~)cosnO+W,~f,~n(~)sinnO
where fm,~(~) and f ~ ( ~ ) are functions of the meridional co-ordinate, 0 is the circumferential angular co-ordinate and in the dynamic case W ~ and W~, are time-dependent coefficients. For the case of homogeneous boundary conditions, i.e. the boundary conditions are independent of the circumferential co-ordinate, there is no coupling between the modes of vibration with different numbers of circumferential waves and the free vibration problem can be solved for each value of n independently. Before these papers came to the author's notice he had been using a similar method for calculating the natural frequencies and normal modes of cylindrical conical and spherical shells and shells made up of combinations of these shapes. In this method the displacements are represented b y truncated power series in the meridional co-ordinate. The coefficients of the terms in the power series and the natural frequencies are found b y finding the stationary values of the Hamiltonian S~(T-V)dt subject to the constraints required to satisfy the boundary conditions at the ends of the shell and the continuity conditions at the junction between shell elements. If the idealization of the shell geometry and the functions representing the displacements are the same in this method as in the finite element method, then the results obtained are identical, even though the formulation of the equations may be different. In most of the published examples used to illustrate the application of ring finite elements the simplest polynomial displacement functions which ensure continuity of displacements and slope at the junctions between elements have been used. In these cases large numbers of elements were required to give satisfactory solutions. In one example, Fig. 6, 8 a reasonable solution for the radial displacement of a cylindrical shell, subjected to a radial ring load, was obtained with a few elements and more than the minimum of terms in the displacement functions. For the dynamic case the author found that good approximations for the fundamental natural frequency and mode were obtained for shells, treated as
F r e e v i b r a t i o n s o f shells o f r e v o l u t i o n u s i n g r i n g finite e l e m e n t s
561
single elements, when only a few terms were used in the power series representing the displacements. These results suggested that extending the polynomials representing the shell displacements would give considerable improvement in the ring finite element method. The object of the work reported here is to show the effect of extending the polynomials representing the displacements of ring finite elements by comparing results obtained for the natural frequencies of some shells with different finite element idealizations. It is difficult to establish a satisfactory basis for comparing different finite elements. A very simple basis, used in this paper, is the number of unknowns in the final equations which have to be solved, i.e. the number of degrees of freedom which the assumed displacement forms allow. The computing time and store required to solve the equations on a digital computer are related to this quantity, but the factors involved for different elements may vary considerably.
ANALYSIS T h e m a s s a n d stiffness m a t r i c e s d e r i v e d b e l o w a r e for t h e n t h h a r m o n i c c o m p o n e n t s of t h e d i s p l a c e m e n t s . I f it is n e c e s s a r y t o i n c l u d e a n u m b e r of h a r m o n i c s t h e c o m p l e t e e l e m e n t m a t r i c e s c a n b e f o r m e d f r o m t h e s u b - m a t r i c e s for e a c h h a r m o n i c ) W h e n t h e r e is n o c o u p l i n g b e t w e e n m o d e s w i t h d i f f e r e n t n u m b e r s of c i r c u m f e r e n t i a l w a v e s t h e free v i b r a t i o n p r o b l e m c a n b e s o l v e d for e a c h h a r m o n i c s e p a r a t e l y .
T r u n c a t e d element
Fig. 1 r e p r e s e n t s a t r u n c a t e d e l e m e n t .
w
\
f FIG. 1. T r u n c a t e d e l e m e n t .
562
J . J . WEBSTER
A s s u m i n g t h a t t h e n t h h a r m o n i c c o m p o n e n t s of t h e mid-surface displacements can be expressed as: u =
~ Um(sfl)~-l(1 or sinn0) = un(1 or sinn0) tnwl M
v = ~. V,~(s/l)"-a(1 or cosn0) = v.(1 or cosn0)
(1)
msl M
w = ~ W~(s/1)m-l(1 or cosn0) = w.(1 or cosn0) The t e r m s (1 or sin nO) a n d (1 or cos nO) allow the a x i s y m m e t r i c ease to be included. C o n t i n u i t y b e t w e e n joining elements is m a i n t a i n e d b y e q u a t i n g nodal displacements and these quantities are expressed in t e r m s of t h e coefficients U~, V~ and W~ from t h e following relationships : q(1) = t a n g e n t i a l displacement = u .
/
q(') = axial displacement
= v . sin ~b- w . cos
q(a~ _- radial displacement
= v . cos ~ + w~ sin ~ -- ~W ~-~n
q(4~ = r o t a t i o n
at s = 0 i
Vn
]|
~
q(SM-a) = t a n g e n t i a l displacement = un
!
q(aM-,)
axial displacement
v . sin ~ -- w~ cos ~b
qCaM-x)
radial displacement
v n cos ~ + w , sin ~b~
at s --- 1
/
qCaM)
Os
rotation
]/
r,
These eight e q u a t i o n s are used to replace eight of t h e coefficients U~, V~ and W~ b y t h e eight nodal displacements, four at each end of t h e element. The first four e q u a t i o n s relate t h e four nodal displacements at 8 = 0 to U1, V1, W1 and W~. The r e m a i n i n g four e q u a t i o n s relate t h e nodal displacements at s = 1 to all the coefficients. These e q u a t i o n s are used to eliminate a f u r t h e r four coefficients, one U~, one V~ and two W~'s. F o r convenience U,, V~, W3 and W4 are eliminated. The r e m a i n i n g elements of the v e c t o r {q} are thus : q('+"~
= U~
m=
q(M+,n)
= V,.
m = 3, 4 . . . . , M
q(,M-4+m) = Wm
m=
3,4 ..... M
5,6 . . . . , M
W e t h u s h a v e the t r a n s f o r m a t i o n : {q} = [B] {x}
(2)
The strain energy in the shell element is
V = ~ ;
Jf-+t /' ,' ' 2( l~v2)E [~, + e~ + 2re, eo + t--2-)/1-v'Y.eJ']sinq~(r.+z)(ra+z)
The strain expressions used are derived from Fliigge s 0v
w
z
[
~'w
v dr,\
eo = r ~ - ~ + ~ c o t ~ \ r a + z / q (re+z)
too
= (r e + z )
r(~u
r,
cot /
. \ 1
+
(reSbz)
/%+z\ r.
1 \
00' + c ° t ~ - ~ s
~v 1
a2w 8sbO
cot sincb ~0)
(3)
F r e e v i b r a t i o n s of shells of r e v o l u t i o n using ring finite elements
563
T h e strain e n e r g y expression, a f t e r s u b s t i t u t i n g for t h e strains, i n t e g r a t i n g t h r o u g h t h e thickness a n d neglecting t e r m s of order > t s, is g i v e n in A p p e n d i x 1. S u b s t i t u t i n g for t h e displacements f r o m e q u a t i o n (1) gives the strain energy V --- ½[x] [K] {x} a n d s u b s t i t u t i n g for {x} f r o m e q u a t i o n (2) gives V ---- {[q] [ B - l ] T [K] [B -a] {q}
[B-X] w [K] [B -t] is t h e stiffness m a t r i x . (The elements of the 3 M x 3 M square m a t r i x [K] are too long to give i n extenso b u t a s t a t e m e n t of t h e m is available on application to t h e a u t h o r ( A p p e n d i x 2).) The kinetic energy, neglecting r o t a t o r y inertia, is: T =
{~2 + 02 + ~b2} _ _ J -t/~
(r, + z) (ra + z) dz (Is dO
(4)
rs
s u b s t i t u t i n g for the velocities f r o m e q u a t i o n (1) and using t h e t r a n s f o r m a t i o n in e q u a t i o n (2) gives T = ½[q] [B-a] T [T] [B -a] {q} [B-a] r [T] [B -L] is t h e e l e m e n t mass m a t r i x a n d t h e elements of the 3 M × 3 M square m a t r i x [T] are also in A p p e n d i x 2. Element with closed end
Fig. 2 represents a shell e l e m e n t w i t h a closed end.
W
S
(31
J Fro. 2. E l e m e n t w i t h closed end.
I f the displacements of the mid-surface of a closed shell e l e m e n t are expressed as polynomials in s t h e n t e r m s of degree less t h a n 2 in s c a n n o t be used as t h e y cause singularities in t h e s t r a i n energy expression. T h e p o w e r series a p p r o x i m a t i o n for t h e displacements m u s t s t a r t a t t h e s ~ t e r m a n d this gives zero displacements a t t h e closed end. This is t h e case for n > 1. I n order to include t h e a x l s y m m e t r i c ease n ---- 0 a n d t h e bending case, n ---- 1, t h e p o w e r series a p p r o x i m a t i o n is a d d e d to four rigid b o d y displacements q~a~, q~2~, qla~ a n d q(4~, shown in Fig. 2. These are t h e displacements a t t h e closed end o f t h e element.
564
J . J . WEBSTER T h e d i s p l a c e m e n t s of t h e m i d - s u r f a c e a r e e x p r e s s e d as :
[
]
u = rq~X)--q~a)-hqC~)+ ~ Urn(s/l) '~+1 (1 or s i n n 0 ) = u~(1 or sin nO) v = [ql2) s i n ~ + q l S ) c o s ~ - q C " s i n a ~ ( r ~ + h ~ ) +
~Vm(s/l) m+l] rtt~l
(1 or c o s n 0 ) = v.(1 or c o s n # ) w =
(5)
-- q(2) cos ¢ + q(3~ sin ~b+ q(4~ cos a ~/(r 2 + h 2) A- ~ Win(s~1)m+l m~l (1 or c o s n 0 ) = w.(1 or cosnO) q(l~ = r o t a t i o n a b o u t axis of s y m m e t r y q(~) = a x i a l d i s p l a c e m e n t qCS~ = r a d i a l d i s p l a c e m e n t q(4) = r o t a t i o n a b o u t axis p e r p e n d i c u l a r to axis of s y m m e t r y qll~ = q[~) = 0
ifn~O
q(S) = q ( 4 ) = 0
ifn¢l
The nodal displacements at the open end are:
q[3M+2~ q(2M+3)
V. sin ~b-- w . cos ~ V. COS ~b+ W. sin ¢ /
qtaM+4)
__ v n / r s + ~Wn/~8
at s = 1
]
T h e r e m a i n i n g e l e m e n t s in t h e v e c t o r {q} a r e : q(S+~)
= U~
m=
2,3 .... ,M
q(M+2+m) = Vm
m = 2, 3 . . . . . M
qt2M+m) = ~',n
m = 3, 4 . . . . . M
We thus have the transformation {q} = [B] {x}
(6)
I n t h i s case t h e first four e l e m e n t s of {x} are t h e rigid b o d y d i s p l a c e m e n t s q(X), qC~, qta~ a n d q(4~ a n d t h e l a s t 3 M e l e m e n t s are t h e coefficients U,., Vm a n d W~. T h e stiffness a n d m a s s m a t r i c e s a r e f o r m e d f r o m t h e e n e r g y expressions (3) a n d (4), t h e d i s p l a c e m e n t f u n c t i o n s , e q u a t i o n (5), a n d t h e t r a n s f o r m a t i o n , e q u a t i o n (6), in t h e s a m e w a y as t h e y were for t h e t r u n c a t e d e l e m e n t . I n t h i s case t h e m a t r i c e s a r e ( 3 M + 4 ) x ( 3 M + 4 ) s q u a r e m a t r i c e s a n d t h e e l e m e n t s of [K] a n d [T] are g i v e n in A p p e n d i x 2 ( a v a i l a b l e f r o m t h e a u t h o r ) . I t s h o u l d b e n o t e d t h a t t h e first four rows a n d c o l u m n s of [K] are n u l l b e c a u s e t h e first four e l e m e n t s of t h e v e c t o r {x} are rigid b o d y d i s p l a c e m e n t s a n d do n o t i n d u c e s t r a i n in t h e shell e l e m e n t .
Computation of element matrices T h e expressions a v a i l a b l e for t h e e l e m e n t s of t h e m a s s a n d stiffness m a t r i c e s i n A p p e n d i x 2 a r e in t h e f o r m of integrals. T h e y a r e a p p l i c a b l e t o shell e l e m e n t s w i t h c o m p l e t e symmetry with regard to geometry and physical properties. The integrals can be evaluated e x p l i c i t l y i n s o m e cases, e.g. for cones, cylinders, p a r t s o f spheres, etc., w i t h u n i f o r m t h i c k n e s s a n d p h y s i c a l p r o p e r t i e s . W h e n t h e explicit e v a l u a t i o n o f t h e i n t e g r a l s is difficult t h e y c a n b e e v a l u a t e d b y n u m e r i c a l i n t e g r a t i o n . I t is t h e n v e r y e a s y t o deal w i t h shells w h o s e g e o m e t r y a n d p h y s i c a l p r o p e r t i e s v a r y a l o n g a m e r i d i a n .
Free vibrations of shells of revolution using ring finite elements
565
Solution for complete shells I t is possible to represent the geometry of any shell of revolution by one or more of the elements described above. The element mass and stiffness matrices are assembled and the boundary conditions specified by manipulating the first and last four rows and columns of the assembled matrices to form the system matrices [S] and [M]. The equation to be solved for the free vibration problem is ( [ ~ ] - A [ M ] ) {~} = 0
The frequency parameter A includes the circular frequency squared, ¢o2, and any other constants t h a t m a y be desired. The frequencies obtained from the solution of this m a t r i x equation, arranged in order of increasing magnitude, are upper bound values for the corresponding true natural frequencies of the shell. E v e n though a particular approximate frequency m a y be much higher than the corresponding true natural frequency, the approximate frequency and its associated mode m a y agree closely with those of a higher frequency mode. NUMERICAL
EXAMPLES
The effect of extending the polynomial displacement functions is illustrated by three examples. I n these examples the natural frequencies have been computed using the analysis given above and treating the shells as single elements. These results are compared with solutions obtained by considering the shells as a number of elements of equal meridional length, using the simplest displacement functions for each element. The simplest displacement functions for the tangential and meridional displacements contain two terms and there are four terms in the simplest normal displacement function. This is the m i n i m u m number of terms required to specify continuity of tangential, meridional and normal displacement and meridional rotation at the junctions between shell elements. An assessment of the accuracy of the finite element solutions m ay be made as exact solutions of the equation of motion are available for these examples.
Simply supported cylindrical shell The usual definition of a simple support for cylindrical shells is used here, i.e. radial displacement, tangential displacement, meridional force and meridional bending moment are zero at the supports. Values of the frequency parameter corresponding to some of the natural frequencies of a simply supported cylindrical shell are given in Table 1. The column headings indicate the method of solution. The exact solutions of the equations of motion were obtained by a method given by Warburton 7. The modes are designated by the three letters u, v and w and an integer. The letters give the relative amplitude of the displacements in order of decreasing magnitude. The omission of one or more letters indicates t h a t these displacements are not present in the mode. The number of meridional half-waves in the displacem e n t form is specified by the integer. For each value of n, the lowest six natural frequencies are tabulated in order of increasing magnitude. An asterisk against a tabulated frequency indicates that the mode associated with this frequency is not the mode specified in the table. The results show that for a given number of degrees of freedom, the natural frequencies of the lower frequency modes are predicted more accurately by the single element representation, with the extended power series displacement functions, than by the multielement representation, with the simplest displacement functions. For the higher frequency modes in which the normal displacement predominates, the multi-element representation sometimes gives more accurate approximations to the natural frequencies. This is to be expected as the proportion of the number of degrees of freedom associated with the normal displacement is greater in this case. In the multi-element representation approximately half the degrees of freedom are associated with the normal displacement, whereas in the single-element method only about a third are associated with the normal displacement. Acceptable accuracies for the natural frequencies of the lower frequency modes are obtained using the single-element representation with only a few terms in the power series
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J . J . WEESTER
r e p r e s e n t i n g t h e d i s p l a c e m e n t s , b u t a r e l a t i v e l y large n u m b e r of t h e e l e m e n t s w i t h t h e simple d i s p l a c e m e n t f u n c t i o n s are r e q u i r e d to o b t a i n s i m i l a r accuracies. TABLE 1.
N A T U R A L F R E Q U E N C Y FACTORS F O R S I M P L Y S U P P O R T E D C Y L I N D R I C A L S H E L L
v = 0.3;
4;
L/R=
0.05
t/R=
A - PR2(1 - v~)¢°~ E
Finite element solutions Single e l e m e n t n
Mode
Multi-e~ment
Exact No. of t e r m s i n p o w e r series 4 6 8
2
No. of e ~ m e n t s 4 5
10
No. of degrees of f r e e d o m
0
8
14
20
8
16
20
40
ul vwl wv2 u2 urv3 wv4
0.2160 0.5047 0-8619 0.8641 0-8984 0.9199
0.2189 0.5050 0-9184 0.9193 1.112" 3.834*
0.2160 0.5047 0.8635 0.8646 0.9066 1.011
0.2160 0.5047 0.8626 0.8641 0.8987 0.9257
0.2627 0.5960 0.8945 1.009" 1.033" 1.156"
0.2273 0.5289 0.8912 0-9297* 0.9425* 1.051"
0.2232 0.5207 0.8842 0.9265* 0.9566* 0.9800*
0.2178 0-5088 0-8689 0.8929 0.9073 0-9334
wuvl wuv2
0.06606 0.3292 0.3501 0.5831 0.7352 0.7891
0.06742 0.3501" 0.3526* 0.7894* 1.951" 2.329*
0.06606 0-3295 0-3501 0.6285 0.7891" 0.8312"
0.06606 0.3292 0.3501 0.5845 0.7404 0.7891
0.09702 0.3501" 0.8502* 0-9100" 1.037" 1.075"
0.07319 0.3501" 0.3933* 0.6868 0.8127" 0.9489*
0.07057 0-3501" 0-3712" 0-6608 0.8051" 0.8250*
0.06716 0.3396 0-3501 0.~041 0.7624 0.7932
0.01475 0.1119 0.2811 0.4580 0.6177 0.7686
0.01529 0.1333 1.400" 1.744" 3.329* 5.478*
0-01475 0.1123 0.3257 0.5691 1.400" 1.743"
0.01475 0.1119 0.2824 0.4645 0.8696 1.380
0.03080 0.9424 1.008 1.052 1.400" 1.902"
0.01801 0-1577 0-3966 0-9701 1-033 1.175
0.01675 0-1411 0-3684 0-5791 1.014 1-093
0'01521 0.1187 0.3033 0.4980 0.6726 0.8364
0-01703 0.05752 0.1543 0.2931 0.4541 0-6307
0-01730 0.07093 3.151" 3-441" 5-175" 10-53"
0.01703 0.05774 0.1884 0.4118 3.151" 3.440*
0.01703 0.05753 0.1554 0.3007 0.7299 1.292
0-02685 0.9807 1-040 1.085 3.151" 3.600*
0.01883 0-08821 0-2610 1.011 1.083 1.244
0.01809 0.07636 0.2305 0.4255 1.064 1.266
0.01725 0.06152 0.1719 0.3336 0-5207 0.7215
v0 wuv3 wuv4
vwul wuvl wuv2 wuv3
wuv4 wuv5
wuv6 wuvl
wuv2 wuv3 wuv4 wuv5 wuv6
* N o t m o d e specified in c o l u m n 2. T h e n u m b e r o f a s t e r i s k s in T a b l e 1 i n d i c a t e s t h a t t h e finite e l e m e n t m e t h o d o f t e n gives a n i n a c c u r a t e p i c t u r e of t h e o r d e r of t h e m o d e s , i.e. w h i c h m o d e s h a v e t h e lowest n a t u r a l frequencies. T h i s is b e c a u s e t h e a c c u r a c y is r e l a t e d t o t h e c o m p l e x i t y of t h e m o d e r a t h e r t h a n t h e m a g n i t u d e of t h e n a t u r a l f r e q u e n c y . E v e n t h o u g h t h e finite e l e m e n t s o l u t i o n m a y b e a v e r y p o o r a p p r o x i m a t i o n t o t h e c o r r e s p o n d i n g e x a c t n a t u r a l f r e q u e n c y a n d m o d e it m a y a g r e e closely w i t h t h a t of a h i g h e r f r e q u e n c y m o d e . F o r e x a m p l e , t h e f o u r t h f r e q u e n c y of t h e 8 degrees of f r e e d o m s i n g l e - e l e m e n t s o l u t i o n , for n = 1, is 0"7894. T h i s f r e q u e n c y a n d its a s s o c i a t e d m o d e are p o o r a p p r o x i m a t i o n s to t h e e x a c t f o u r t h m o d e , i.e. t h e wuv3 m o d e , b u t t h e y a g r e e v e r y well w i t h t h e e x a c t s i x t h m o d e , i.e. v w u l . M a n y similar e x a m p l e s m a y b e f o u n d in T a b l e 1.
F r e e v i b r a t i o n s o f shells o f r e v o l u t i o n using ring finite e l e m e n t s Flat circular plate, clamped
567
at outside edge
T h e l o w e s t six n a t u r a l f r e q u e n c i e s o f a flat circular p l a t e , c l a m p e d a t t h e o u t s i d e edge, for e a c h v a l u e o f n = 0, 1, 2 a n d 3 are g i v e n in T a b l e 2. T h e e x a c t solutions o f t h e e q u a t i o n s o f m o t i o n are g i v e n b y K a l n i n s 8. T h e n o t a t i o n u s e d for defining t h e t y p e o f m o d e is t h e s a m e as t h a t u s e d for t h e c y l i n d e r in T a b l e 1. T A B L E 2.
N O N - D I M E N S I O N A L N A T U R A L F R E Q U E N C I E S OF FLAT CIRCULAR PLATE~ CLAMPED AT O U T S I D E E D G E
= coR(p/E)i;
v = 0.3;
t / R = 0.1
F i n i t e e l e m e n t solutions Single e l e m e n t Mode type
Exact
Multi-element
No. o f t e r m s in p o w e r series 4 6 7 10
16
3
No. o f e l e m e n t s 5
No. o f d e g r e e s o f f r e e d o m 19 10
6
18
22
w w u w v u
0.3091 1.204 2"376 2-696 4.017 4-351
0.3091 1.220 2.376 2.816 4.064 4-353
0.3091 1.204 2.376 2-699 4.032 4.351
0.3091 1.204 2.376 2.699 4.026 4.351
0.3092 1.210 2.434 2.746 4-334 4-684
0.3091 1.204 2.401 2-710 4.122 4.570
0-3091 1-204 2-394 2.703 4-088 4-511
w w
0.6434 1"841 2.052 3-332 3.632 5"250
0.6437 1-889 2.052 3"333 4-709 5-295
0-6434 1.842 2.052 3.332 3.686 5.250
0.6434 1.841 2.052 3.332 3.643 5.250
0.6440 1.858 2.060 3.523 3.980 5-898
0.6435 1.844 2.056 3-405 3.664 5-606
0.6434 1.842 2.055 3.385 3-650 5"513
vu uv
w vu
No. o f degrees o f f r e e d o m
w w vu uv
w vu
w w vu uv
w vu
8
14
17
8
16
20
1-056 2.560 3-195 4-283 4"655 6.142
1.064 2-668 3-235 4-315 6.992* 7-327*
1.055 2-561 3.209 4.294 5.033 6.175
1.055 2.561 3.203 4.290 4.669 6-172
1.059 2.592 3.531 4-889 5.348 7.694
1.056 2-568 3.312 4-509 4.710 7.177
1-056 2.564 3.275 4.438 4-684 6.871
1"548 3"360 4"154 5.266 5"759 7"011
1.554 4-162" 4.473* 5.295 7.498* 8-665*
1.544 3-405 4-154 5.267 6.232 7.036
1.544 3.364 4.154 5-266 5-947 7.014
1.553 3.526 4.428 5.879 6.866 9.111
1.545 3.375 4-250
1.545 3.367 4.222 5.427 5.804 6-745
5.495
5.833 7.944
* N o t m o d e t y p e specified in c o l u m n 2. C o m p a r i n g t h e single e l e m e n t a n d m u l t i - e l e m e n t solutions for a p p r o x i m a t e l y e q u a l n u m b e r s of degrees o f f r e e d o m , it m a y be seen t h a t for t h e t r a n s v e r s e m o d e s , i.e. t h e m o d e s in w h i c h n o r m a l d i s p l a c e m e n t s o n l y are p r e s e n t , t h e r e is little t o choose b e t w e e n t h e t w o e l e m e n t s . F r o m t h i s it is c o n c l u d e d t h a t a n i m p r o v e m e n t in t h e r e p r e s e n t a t i o n o f t h e n o r m a l d i s p l a c e m e n t w i t h a h i g h degree p o l y n o m i a l in t h e s i n g l e - e l e m e n t m e t h o d , r a t h e r t h a n b y piecewise r e p r e s e n t a t i o n w i t h a cubic as is u s e d in t h e m u l t i - e l e m e n t m e t h o d , is offset b y t h e r e d u c t i o n in t h e n u m b e r of degrees o f f r e e d o m a s s o c i a t e d w i t h t h e n o r m a l
568
J.J.
WEBSTER
d i s p l a c e m e n t . T h e s i n g l e - e l e m e n t s o l u t i o n s for t h e i n - p l a n e m o d e s a r e m o r e a c c u r a t e t h a n t h e m u l t i - e l e m e n t s o l u t i o n s b e c a u s e t h e r e is b e t t e r r e p r e s e n t a t i o n of t h e d i s p l a c e m e n t s a n d t h e r e a r e m o r e degrees of f r e e d o m a s s o c i a t e d w i t h t h e i n - p l a n e d i s p l a c e m e n t s i n t h e singleelement method. T h e difference b e t w e e n r e p r e s e n t i n g t h e n o r m a l d i s p l a c e m e n t of t h e whole p l a t e b y a single p o l y n o m i a l or piecewise b y a n u m b e r of cubics, w h e n t h e n u m b e r of degrees of f r e e d o m a s s o c i a t e d w i t h t h i s d i s p l a c e m e n t is t h e s a m e in b o t h cases, m a y b e seen b y c o m p a r i n g t h e f r e q u e n c i e s o f t h e t r a n s v e r s e m o d e s for t h e s i n g l e - e l e m e n t 16 degrees of f r e e d o m s o l u t i o n w i t h t h e 3 - e l e m e n t 10 degrees of f r e e d o m solution. T h e s o l u t i o n s for t h e t r a n s v e r s e m o d e s are i n d e p e n d e n t of t h e r e p r e s e n t a t i o n of t h e i n - p l a n e d i s p l a c e m e n t s b e c a u s e t h e s e d i s p l a c e m e n t s a r e n o t p r e s e n t i n t h e s e m o d e s . T h e n a t u r a l f r e q u e n c i e s for all t h e t r a n s v e r s e m o d e s are p r e d i c t e d m o r e a c c u r a t e l y b y t h e 6 - t e r m p o l y n o m i a l plus t w o rigid b o d y d i s p l a c e m e n t s t h a n b y piecewise r e p r e s e n t a t i o n of t h e n o r m a l d i s p l a c e m e n t b y t h r e e cubics. Spherical
shell
T h r e e s o l u t i o n s for t h e e l e v e n lowest n a t u r a l f r e q u e n c i e s of t h e a x i s y n u n e t r i c m o d e s of a s p h e r i c a l shell, closed a t o n e e n d a n d w i t h t h e edge free or c l a m p e d , a r e g i v e n i n T a b l e 3. T h e e x a c t s o l u t i o n s are g i v e n b y K a l n i n s 9 a n d t h e m u l t i - e l e m e n t r e s u l t s b y N a v a r a t n a 5. T h e s i n g l e - e l e m e n t r e s u l t s were o b t a i n e d b y t h e m e t h o d g i v e n a b o v e . K a l n i n s '9 classificat i o n for t h e b e n d i n g a n d m e m b r a n e or s t r e t c h i n g m o d e s is used. I n t h e s e m o d e s o n l y n o r m a l a n d m e r i d i o n a l d i s p l a c e m e n t s a r e p r e s e n t a n d in t h e t o r s i o n a l m o d e s t h e r e is only tangential motion. TABLE 3. NON-DIMENSIONAL NATURAL FREQUENCIES OF SPHERICAL SHELL v -~ 0"3;
= wR(p/E)t;
Mode No.
Exact (Ref. 9)
t / R = 0"05;
Multielement (Ref. 5)
h a l f - a n g l e --- 60 °
Single element
Modet type
0.931 1.089 1.535 2.360 2.551 2.985 4.023 4.950 5.548* 6.224* 6.862*
B B B B S T B T B S B
1.007 1.391 1.700 2.095 2.386 3.851 4.062* 4.151" 5.962* 6.208* 7.148"
B B B T B B S T B T B
F r e e edge 1 2 3 4 5 6 7 8 9 10 lI
0.931 1.088 1.533 2.348 2.544 -3.497 -4.951 5.230 6.693
0.932 1.094 1.544 2.363 2.548 2.982 3.519 4.971 4.980 5.543 6-734 C l a m p e d edge
1 2 3 4 5 6 7 8 9 10 11
1.006 1.391 --2.375 3.486 3.991 -4.947 --
6.690
1-008 1.395 1.702 2.126 2.387 3.506 3.996 4.159 5.001 6.037 6.729
* N o t m o d e t y p e specified. t B--bending, S--stretching and T--torsional.
Free vibrations of shells of revolution using ring finite elements
569
For the multi-element solution Navaratna represented the shell by 24 curved elements of equal meridional length. The displacement functions are the simplest ones required to maintain continuity between joining elements and the number of degrees of freedom was 98 for the free edge shell and 94 for the clamped edge case. In the single element solution the displacements were represented by four rigid body displacements and three 7-term polynomials. After eliminating the co-ordinates required to satisfy the boundary conditions there are 23 degrees of freedom for the free shell and 19 for the clamped edge shell. Comparing the two finite-element solutions, it may be seen that the single-element solutions are better for the lower frequency modes even though the single-element representation has considerably fewer degrees of freedom than the multi-element representation. CONCLUSIONS T h e results for t h e n a t u r a l frequencies for t h e t h r e e shells g i v e n a b o v e indicate t h a t t h e r e are considerable a d v a n t a g e s in r e p r e s e n t i n g a shell b y a few e l e m e n t s w i t h d i s p l a c e m e n t functions e x t e n d e d b e y o n d t h a t required to satisfy c o n t i n u i t y b e t w e e n e l e m e n t s r a t h e r t h a n using a large n u m b e r of e l e m e n t s w i t h t h e s i m p l e s t d i s p l a c e m e n t functions. F o r a g i v e n n u m b e r o f degrees o f f r e e d o m , usually m o r e a c c u r a t e results are o b t a i n e d b y e x t e n d i n g t h e d i s p l a c e m e n t functions r a t h e r t h a n b y using m o r e e l e m e n t s w i t h simpler d i s p l a c e m e n t functions. Generally, in m o d e s o f v i b r a t i o n in which m o r e t h a n one d i s p l a c e m e n t is present, t h e f o r m o f t h e d i s p l a c e m e n t s is similar, i.e. t h e y h a v e similar n u m b e r s of nodes. F o r this r e a s o n p o l y n o m i a l s o f t h e s a m e degree are used to r e p r e s e n t all t h r e e displacements. I f m o r e t h a n a b o u t s e v e n t e r m s are used in t h e displacem e n t functions t h e e q u a t i o n s m a y b e c o m e ill c o n d i t i o n e d a n d difficult to solve a c c u r a t e l y . U s u a l l y d i s p l a c e m e n t functions of this l e n g t h give a d e q u a t e a c c u r a c y for t h e lower f r e q u e n c y modes. I f t h e higher f r e q u e n c y m o d e s are of i n t e r e s t m u l t i - l e n g t h a r i t h m e t i c could be used to solve t h e equations, b u t it would p r o b a b l y be m o r e c o n v e n i e n t to r e p r e s e n t t h e shell b y m o r e t h a n one element. A n o t h e r r e a s o n for using m o r e t h a n one e l e m e n t is t h a t t h e d i s p l a c e m e n t functions are c o n t i n u o u s w i t h i n t h e e l e m e n t s a n d c a n n o t r e p r e s e n t e x a c t l y discontinuities in d i s p l a c e m e n t or d e r i v a t i v e s of t h e displacements. I t would s e e m to be a d v i s a b l e to divide shells w i t h discontinuities in g e o m e t r y or physical p r o p e r t i e s into a n u m b e r of e l e m e n t s a t t h e points of discontinuity. Acknowledgement--The author gratefully acknowledges the assistance and advice given
by Professor G. B. Warburton during the course of this work. REFERENCES I. P. E. GR~TON and D. R. STRO~m, A.I.A.A. Jl I, 2343 (1963). 2. E. P. PoPov, J. PENZ~N and Z. A. Lu, Jl Eng. Mech. Div., Proc. Am. ,~oc. cir. Engrs 96,
119 (1964). 3. J. H. PERCY, T. H. H. PIAN, S. KL~.IN and D. R. NAV~A~A, A . I . A . A . Jl 3, 2138(1965). 4. R. E. JONES and D. R. STRO~, A . I . A . A . J l 4, 1519 (1966). 5. D. R. N A V ~ A T ~ A , A . I . A . A . Jl 4, 2056 (1966). 6. W. FLi2GGE, Stresses in Shells, p. 317. Springer-Verlag, Berlin (1962). 7. G. B. WAa~BVRTON,J. Mech. Engng Sci. 7, 399 (1965). 8. A. K_~Nr~s, Prec. 4th U.S. hath. Congr. appl. Mech. 1, 225 (1962). 9. A. K ~ i ~ s , J. acoust. See. A m . 36, 74 (1964). 38
+ 2(1-v)/O~w r [~-~
w~
r
00 ~
,,
~
v Ov~
(10'w
cos~ ) 24 (1-v)sin 4 8v - - ~Ow ~ ( Ou 8 ,, cos ¢ ) r rr,
(e -
sin4]
usha 4 ~u vvcos 4
w + ~ ( r , -~~'w + ~, ~,d,-,~, a~!
sin¢] wWSin¢ (1
Ov [sha4 r]
4~+CZ--.-~f
1
_2(~r, Ou vcos 4(2
-~
2(s"
t~ [sin4/ov\~
Strain energy in shell element
APPENDIX 1
+cos¢
Ou))/
O'w v dr.] sinc~lOu\'
r - tOO!
F R E E V I B R A T I O N S OF SHELLS OF R E V O L U T I O N USING R I N G F I N I T E ELEMENTS J. J. WEBSTER D e p a r t m e n t of Mechanical Engineering, University of Nottingham
(Received 9 February 1967) S u m m a r y - - I m p r o v e m e n t s in the ring finite element analysis, for shells of revolution, obtained b y extending the polynomials representing the displacements are illustrated b y comparing the natural frequencies of three shells obtained from this improved analysis with those obtained b y the usual method. The mass and stiffness matrices and their derivation for the elements with extended displacement functions are given.
h l n
{q} r rs~ r e 8
t U~ V, '/0 Un~ Vn~ Wn
{x} Z
[B] E
[K] R T
IT]
U,.,V,.,W,.
V Ot
0
P
~k 03
q~ k(i,j) [B-l] [B-~]~"
NOTATION axial distance from apex of element with closed end, see Fig. 2 meridional length of element number of cirenmferential waves column vector of generalized co-ordinates radius principal radii of curvature meridional co-ordinate shell thickness tangential, meridional and normal displacements amplitudes of the n t h harmonic components of the displacements defined in equation (1) column vector of displacement function coefficients normal co-ordinate, through shell thickness transformation m a t r i x relating {q} to {x} Young's modulus square m a t r i x in strain energy expression radius kinetic energy square m a t r i x in kinetic energy expression coefficients of ruth terms in displacement function strain energy angle, defined in Fig. 2 meridional, tangential and shear strains circumferential co-ordinate Poisson's ratio density angle between normal to shell surface and axis of shell circular frequency i t h element of column vector {q} j t h element of i t h row of m a t r i x [K] inverse of m a t r i x [B] transpose of inverse of [B] INTRODUCTION
FINITE element methods are used extensively to solve both static and dynamic problems for elastic structures.
For shells of revolution the structure has been 559
560
J . J . W~.BS~R
idealized either b y fiat plate elements or more usually by ring elements bounded b y surfaces of constant meridional co-ordinate. This paper is concerned with the latter approach. In the earlier work (Refs. 1, 2, 3) truncated cone elements were used, even for shells with meridional curvature. Large numbers of these elements were sometimes required to give a reasonable approximation to the shell geometry and recently 4,5 elements with meridional curvature have been developed. Exact expressions for the influence coefficients for some elements subjected to axisymmetric edge loads have been derived, 2 b u t in general the displacements of the middle surface of the elements are represented by a number of functions of the meridional co-ordinate. The coefficients of these functions, and in the free vibration problem the natural frequencies, are in effect determined by finding their values which make the relevant energy integral stationary. A typical displacement, say the normal displacement w, is represented by:
w= •
ofmo(~)+ [Wm,Jm~(~)cosnO+W,~f,~n(~)sinnO
where fm,~(~) and f ~ ( ~ ) are functions of the meridional co-ordinate, 0 is the circumferential angular co-ordinate and in the dynamic case W ~ and W~, are time-dependent coefficients. For the case of homogeneous boundary conditions, i.e. the boundary conditions are independent of the circumferential co-ordinate, there is no coupling between the modes of vibration with different numbers of circumferential waves and the free vibration problem can be solved for each value of n independently. Before these papers came to the author's notice he had been using a similar method for calculating the natural frequencies and normal modes of cylindrical conical and spherical shells and shells made up of combinations of these shapes. In this method the displacements are represented b y truncated power series in the meridional co-ordinate. The coefficients of the terms in the power series and the natural frequencies are found b y finding the stationary values of the Hamiltonian S~(T-V)dt subject to the constraints required to satisfy the boundary conditions at the ends of the shell and the continuity conditions at the junction between shell elements. If the idealization of the shell geometry and the functions representing the displacements are the same in this method as in the finite element method, then the results obtained are identical, even though the formulation of the equations may be different. In most of the published examples used to illustrate the application of ring finite elements the simplest polynomial displacement functions which ensure continuity of displacements and slope at the junctions between elements have been used. In these cases large numbers of elements were required to give satisfactory solutions. In one example, Fig. 6, 8 a reasonable solution for the radial displacement of a cylindrical shell, subjected to a radial ring load, was obtained with a few elements and more than the minimum of terms in the displacement functions. For the dynamic case the author found that good approximations for the fundamental natural frequency and mode were obtained for shells, treated as
F r e e v i b r a t i o n s o f shells o f r e v o l u t i o n u s i n g r i n g finite e l e m e n t s
561
single elements, when only a few terms were used in the power series representing the displacements. These results suggested that extending the polynomials representing the shell displacements would give considerable improvement in the ring finite element method. The object of the work reported here is to show the effect of extending the polynomials representing the displacements of ring finite elements by comparing results obtained for the natural frequencies of some shells with different finite element idealizations. It is difficult to establish a satisfactory basis for comparing different finite elements. A very simple basis, used in this paper, is the number of unknowns in the final equations which have to be solved, i.e. the number of degrees of freedom which the assumed displacement forms allow. The computing time and store required to solve the equations on a digital computer are related to this quantity, but the factors involved for different elements may vary considerably.
ANALYSIS T h e m a s s a n d stiffness m a t r i c e s d e r i v e d b e l o w a r e for t h e n t h h a r m o n i c c o m p o n e n t s of t h e d i s p l a c e m e n t s . I f it is n e c e s s a r y t o i n c l u d e a n u m b e r of h a r m o n i c s t h e c o m p l e t e e l e m e n t m a t r i c e s c a n b e f o r m e d f r o m t h e s u b - m a t r i c e s for e a c h h a r m o n i c ) W h e n t h e r e is n o c o u p l i n g b e t w e e n m o d e s w i t h d i f f e r e n t n u m b e r s of c i r c u m f e r e n t i a l w a v e s t h e free v i b r a t i o n p r o b l e m c a n b e s o l v e d for e a c h h a r m o n i c s e p a r a t e l y .
T r u n c a t e d element
Fig. 1 r e p r e s e n t s a t r u n c a t e d e l e m e n t .
w
\
f FIG. 1. T r u n c a t e d e l e m e n t .
562
J . J . WEBSTER
A s s u m i n g t h a t t h e n t h h a r m o n i c c o m p o n e n t s of t h e mid-surface displacements can be expressed as: u =
~ Um(sfl)~-l(1 or sinn0) = un(1 or sinn0) tnwl M
v = ~. V,~(s/l)"-a(1 or cosn0) = v.(1 or cosn0)
(1)
msl M
w = ~ W~(s/1)m-l(1 or cosn0) = w.(1 or cosn0) The t e r m s (1 or sin nO) a n d (1 or cos nO) allow the a x i s y m m e t r i c ease to be included. C o n t i n u i t y b e t w e e n joining elements is m a i n t a i n e d b y e q u a t i n g nodal displacements and these quantities are expressed in t e r m s of t h e coefficients U~, V~ and W~ from t h e following relationships : q(1) = t a n g e n t i a l displacement = u .
/
q(') = axial displacement
= v . sin ~b- w . cos
q(a~ _- radial displacement
= v . cos ~ + w~ sin ~ -- ~W ~-~n
q(4~ = r o t a t i o n
at s = 0 i
Vn
]|
~
q(SM-a) = t a n g e n t i a l displacement = un
!
q(aM-,)
axial displacement
v . sin ~ -- w~ cos ~b
qCaM-x)
radial displacement
v n cos ~ + w , sin ~b~
at s --- 1
/
qCaM)
Os
rotation
]/
r,
These eight e q u a t i o n s are used to replace eight of t h e coefficients U~, V~ and W~ b y t h e eight nodal displacements, four at each end of t h e element. The first four e q u a t i o n s relate t h e four nodal displacements at 8 = 0 to U1, V1, W1 and W~. The r e m a i n i n g four e q u a t i o n s relate t h e nodal displacements at s = 1 to all the coefficients. These e q u a t i o n s are used to eliminate a f u r t h e r four coefficients, one U~, one V~ and two W~'s. F o r convenience U,, V~, W3 and W4 are eliminated. The r e m a i n i n g elements of the v e c t o r {q} are thus : q('+"~
= U~
m=
q(M+,n)
= V,.
m = 3, 4 . . . . , M
q(,M-4+m) = Wm
m=
3,4 ..... M
5,6 . . . . , M
W e t h u s h a v e the t r a n s f o r m a t i o n : {q} = [B] {x}
(2)
The strain energy in the shell element is
V = ~ ;
Jf-+t /' ,' ' 2( l~v2)E [~, + e~ + 2re, eo + t--2-)/1-v'Y.eJ']sinq~(r.+z)(ra+z)
The strain expressions used are derived from Fliigge s 0v
w
z
[
~'w
v dr,\
eo = r ~ - ~ + ~ c o t ~ \ r a + z / q (re+z)
too
= (r e + z )
r(~u
r,
cot /
. \ 1
+
(reSbz)
/%+z\ r.
1 \
00' + c ° t ~ - ~ s
~v 1
a2w 8sbO
cot sincb ~0)
(3)
F r e e v i b r a t i o n s of shells of r e v o l u t i o n using ring finite elements
563
T h e strain e n e r g y expression, a f t e r s u b s t i t u t i n g for t h e strains, i n t e g r a t i n g t h r o u g h t h e thickness a n d neglecting t e r m s of order > t s, is g i v e n in A p p e n d i x 1. S u b s t i t u t i n g for t h e displacements f r o m e q u a t i o n (1) gives the strain energy V --- ½[x] [K] {x} a n d s u b s t i t u t i n g for {x} f r o m e q u a t i o n (2) gives V ---- {[q] [ B - l ] T [K] [B -a] {q}
[B-X] w [K] [B -t] is t h e stiffness m a t r i x . (The elements of the 3 M x 3 M square m a t r i x [K] are too long to give i n extenso b u t a s t a t e m e n t of t h e m is available on application to t h e a u t h o r ( A p p e n d i x 2).) The kinetic energy, neglecting r o t a t o r y inertia, is: T =
{~2 + 02 + ~b2} _ _ J -t/~
(r, + z) (ra + z) dz (Is dO
(4)
rs
s u b s t i t u t i n g for the velocities f r o m e q u a t i o n (1) and using t h e t r a n s f o r m a t i o n in e q u a t i o n (2) gives T = ½[q] [B-a] T [T] [B -a] {q} [B-a] r [T] [B -L] is t h e e l e m e n t mass m a t r i x a n d t h e elements of the 3 M × 3 M square m a t r i x [T] are also in A p p e n d i x 2. Element with closed end
Fig. 2 represents a shell e l e m e n t w i t h a closed end.
W
S
(31
J Fro. 2. E l e m e n t w i t h closed end.
I f the displacements of the mid-surface of a closed shell e l e m e n t are expressed as polynomials in s t h e n t e r m s of degree less t h a n 2 in s c a n n o t be used as t h e y cause singularities in t h e s t r a i n energy expression. T h e p o w e r series a p p r o x i m a t i o n for t h e displacements m u s t s t a r t a t t h e s ~ t e r m a n d this gives zero displacements a t t h e closed end. This is t h e case for n > 1. I n order to include t h e a x l s y m m e t r i c ease n ---- 0 a n d t h e bending case, n ---- 1, t h e p o w e r series a p p r o x i m a t i o n is a d d e d to four rigid b o d y displacements q~a~, q~2~, qla~ a n d q(4~, shown in Fig. 2. These are t h e displacements a t t h e closed end o f t h e element.
564
J . J . WEBSTER T h e d i s p l a c e m e n t s of t h e m i d - s u r f a c e a r e e x p r e s s e d as :
[
]
u = rq~X)--q~a)-hqC~)+ ~ Urn(s/l) '~+1 (1 or s i n n 0 ) = u~(1 or sin nO) v = [ql2) s i n ~ + q l S ) c o s ~ - q C " s i n a ~ ( r ~ + h ~ ) +
~Vm(s/l) m+l] rtt~l
(1 or c o s n 0 ) = v.(1 or c o s n # ) w =
(5)
-- q(2) cos ¢ + q(3~ sin ~b+ q(4~ cos a ~/(r 2 + h 2) A- ~ Win(s~1)m+l m~l (1 or c o s n 0 ) = w.(1 or cosnO) q(l~ = r o t a t i o n a b o u t axis of s y m m e t r y q(~) = a x i a l d i s p l a c e m e n t qCS~ = r a d i a l d i s p l a c e m e n t q(4) = r o t a t i o n a b o u t axis p e r p e n d i c u l a r to axis of s y m m e t r y qll~ = q[~) = 0
ifn~O
q(S) = q ( 4 ) = 0
ifn¢l
The nodal displacements at the open end are:
q[3M+2~ q(2M+3)
V. sin ~b-- w . cos ~ V. COS ~b+ W. sin ¢ /
qtaM+4)
__ v n / r s + ~Wn/~8
at s = 1
]
T h e r e m a i n i n g e l e m e n t s in t h e v e c t o r {q} a r e : q(S+~)
= U~
m=
2,3 .... ,M
q(M+2+m) = Vm
m = 2, 3 . . . . . M
qt2M+m) = ~',n
m = 3, 4 . . . . . M
We thus have the transformation {q} = [B] {x}
(6)
I n t h i s case t h e first four e l e m e n t s of {x} are t h e rigid b o d y d i s p l a c e m e n t s q(X), qC~, qta~ a n d q(4~ a n d t h e l a s t 3 M e l e m e n t s are t h e coefficients U,., Vm a n d W~. T h e stiffness a n d m a s s m a t r i c e s a r e f o r m e d f r o m t h e e n e r g y expressions (3) a n d (4), t h e d i s p l a c e m e n t f u n c t i o n s , e q u a t i o n (5), a n d t h e t r a n s f o r m a t i o n , e q u a t i o n (6), in t h e s a m e w a y as t h e y were for t h e t r u n c a t e d e l e m e n t . I n t h i s case t h e m a t r i c e s a r e ( 3 M + 4 ) x ( 3 M + 4 ) s q u a r e m a t r i c e s a n d t h e e l e m e n t s of [K] a n d [T] are g i v e n in A p p e n d i x 2 ( a v a i l a b l e f r o m t h e a u t h o r ) . I t s h o u l d b e n o t e d t h a t t h e first four rows a n d c o l u m n s of [K] are n u l l b e c a u s e t h e first four e l e m e n t s of t h e v e c t o r {x} are rigid b o d y d i s p l a c e m e n t s a n d do n o t i n d u c e s t r a i n in t h e shell e l e m e n t .
Computation of element matrices T h e expressions a v a i l a b l e for t h e e l e m e n t s of t h e m a s s a n d stiffness m a t r i c e s i n A p p e n d i x 2 a r e in t h e f o r m of integrals. T h e y a r e a p p l i c a b l e t o shell e l e m e n t s w i t h c o m p l e t e symmetry with regard to geometry and physical properties. The integrals can be evaluated e x p l i c i t l y i n s o m e cases, e.g. for cones, cylinders, p a r t s o f spheres, etc., w i t h u n i f o r m t h i c k n e s s a n d p h y s i c a l p r o p e r t i e s . W h e n t h e explicit e v a l u a t i o n o f t h e i n t e g r a l s is difficult t h e y c a n b e e v a l u a t e d b y n u m e r i c a l i n t e g r a t i o n . I t is t h e n v e r y e a s y t o deal w i t h shells w h o s e g e o m e t r y a n d p h y s i c a l p r o p e r t i e s v a r y a l o n g a m e r i d i a n .
Free vibrations of shells of revolution using ring finite elements
565
Solution for complete shells I t is possible to represent the geometry of any shell of revolution by one or more of the elements described above. The element mass and stiffness matrices are assembled and the boundary conditions specified by manipulating the first and last four rows and columns of the assembled matrices to form the system matrices [S] and [M]. The equation to be solved for the free vibration problem is ( [ ~ ] - A [ M ] ) {~} = 0
The frequency parameter A includes the circular frequency squared, ¢o2, and any other constants t h a t m a y be desired. The frequencies obtained from the solution of this m a t r i x equation, arranged in order of increasing magnitude, are upper bound values for the corresponding true natural frequencies of the shell. E v e n though a particular approximate frequency m a y be much higher than the corresponding true natural frequency, the approximate frequency and its associated mode m a y agree closely with those of a higher frequency mode. NUMERICAL
EXAMPLES
The effect of extending the polynomial displacement functions is illustrated by three examples. I n these examples the natural frequencies have been computed using the analysis given above and treating the shells as single elements. These results are compared with solutions obtained by considering the shells as a number of elements of equal meridional length, using the simplest displacement functions for each element. The simplest displacement functions for the tangential and meridional displacements contain two terms and there are four terms in the simplest normal displacement function. This is the m i n i m u m number of terms required to specify continuity of tangential, meridional and normal displacement and meridional rotation at the junctions between shell elements. An assessment of the accuracy of the finite element solutions m ay be made as exact solutions of the equation of motion are available for these examples.
Simply supported cylindrical shell The usual definition of a simple support for cylindrical shells is used here, i.e. radial displacement, tangential displacement, meridional force and meridional bending moment are zero at the supports. Values of the frequency parameter corresponding to some of the natural frequencies of a simply supported cylindrical shell are given in Table 1. The column headings indicate the method of solution. The exact solutions of the equations of motion were obtained by a method given by Warburton 7. The modes are designated by the three letters u, v and w and an integer. The letters give the relative amplitude of the displacements in order of decreasing magnitude. The omission of one or more letters indicates t h a t these displacements are not present in the mode. The number of meridional half-waves in the displacem e n t form is specified by the integer. For each value of n, the lowest six natural frequencies are tabulated in order of increasing magnitude. An asterisk against a tabulated frequency indicates that the mode associated with this frequency is not the mode specified in the table. The results show that for a given number of degrees of freedom, the natural frequencies of the lower frequency modes are predicted more accurately by the single element representation, with the extended power series displacement functions, than by the multielement representation, with the simplest displacement functions. For the higher frequency modes in which the normal displacement predominates, the multi-element representation sometimes gives more accurate approximations to the natural frequencies. This is to be expected as the proportion of the number of degrees of freedom associated with the normal displacement is greater in this case. In the multi-element representation approximately half the degrees of freedom are associated with the normal displacement, whereas in the single-element method only about a third are associated with the normal displacement. Acceptable accuracies for the natural frequencies of the lower frequency modes are obtained using the single-element representation with only a few terms in the power series
566
J . J . WEESTER
r e p r e s e n t i n g t h e d i s p l a c e m e n t s , b u t a r e l a t i v e l y large n u m b e r of t h e e l e m e n t s w i t h t h e simple d i s p l a c e m e n t f u n c t i o n s are r e q u i r e d to o b t a i n s i m i l a r accuracies. TABLE 1.
N A T U R A L F R E Q U E N C Y FACTORS F O R S I M P L Y S U P P O R T E D C Y L I N D R I C A L S H E L L
v = 0.3;
4;
L/R=
0.05
t/R=
A - PR2(1 - v~)¢°~ E
Finite element solutions Single e l e m e n t n
Mode
Multi-e~ment
Exact No. of t e r m s i n p o w e r series 4 6 8
2
No. of e ~ m e n t s 4 5
10
No. of degrees of f r e e d o m
0
8
14
20
8
16
20
40
ul vwl wv2 u2 urv3 wv4
0.2160 0.5047 0-8619 0.8641 0-8984 0.9199
0.2189 0.5050 0-9184 0.9193 1.112" 3.834*
0.2160 0.5047 0.8635 0.8646 0.9066 1.011
0.2160 0.5047 0.8626 0.8641 0.8987 0.9257
0.2627 0.5960 0.8945 1.009" 1.033" 1.156"
0.2273 0.5289 0.8912 0-9297* 0.9425* 1.051"
0.2232 0.5207 0.8842 0.9265* 0.9566* 0.9800*
0.2178 0-5088 0-8689 0.8929 0.9073 0-9334
wuvl wuv2
0.06606 0.3292 0.3501 0.5831 0.7352 0.7891
0.06742 0.3501" 0.3526* 0.7894* 1.951" 2.329*
0.06606 0-3295 0-3501 0.6285 0.7891" 0.8312"
0.06606 0.3292 0.3501 0.5845 0.7404 0.7891
0.09702 0.3501" 0.8502* 0-9100" 1.037" 1.075"
0.07319 0.3501" 0.3933* 0.6868 0.8127" 0.9489*
0.07057 0-3501" 0-3712" 0-6608 0.8051" 0.8250*
0.06716 0.3396 0-3501 0.~041 0.7624 0.7932
0.01475 0.1119 0.2811 0.4580 0.6177 0.7686
0.01529 0.1333 1.400" 1.744" 3.329* 5.478*
0-01475 0.1123 0.3257 0.5691 1.400" 1.743"
0.01475 0.1119 0.2824 0.4645 0.8696 1.380
0.03080 0.9424 1.008 1.052 1.400" 1.902"
0.01801 0-1577 0-3966 0-9701 1-033 1.175
0.01675 0-1411 0-3684 0-5791 1.014 1-093
0'01521 0.1187 0.3033 0.4980 0.6726 0.8364
0-01703 0.05752 0.1543 0.2931 0.4541 0-6307
0-01730 0.07093 3.151" 3-441" 5-175" 10-53"
0.01703 0.05774 0.1884 0.4118 3.151" 3.440*
0.01703 0.05753 0.1554 0.3007 0.7299 1.292
0-02685 0.9807 1-040 1.085 3.151" 3.600*
0.01883 0-08821 0-2610 1.011 1.083 1.244
0.01809 0.07636 0.2305 0.4255 1.064 1.266
0.01725 0.06152 0.1719 0.3336 0-5207 0.7215
v0 wuv3 wuv4
vwul wuvl wuv2 wuv3
wuv4 wuv5
wuv6 wuvl
wuv2 wuv3 wuv4 wuv5 wuv6
* N o t m o d e specified in c o l u m n 2. T h e n u m b e r o f a s t e r i s k s in T a b l e 1 i n d i c a t e s t h a t t h e finite e l e m e n t m e t h o d o f t e n gives a n i n a c c u r a t e p i c t u r e of t h e o r d e r of t h e m o d e s , i.e. w h i c h m o d e s h a v e t h e lowest n a t u r a l frequencies. T h i s is b e c a u s e t h e a c c u r a c y is r e l a t e d t o t h e c o m p l e x i t y of t h e m o d e r a t h e r t h a n t h e m a g n i t u d e of t h e n a t u r a l f r e q u e n c y . E v e n t h o u g h t h e finite e l e m e n t s o l u t i o n m a y b e a v e r y p o o r a p p r o x i m a t i o n t o t h e c o r r e s p o n d i n g e x a c t n a t u r a l f r e q u e n c y a n d m o d e it m a y a g r e e closely w i t h t h a t of a h i g h e r f r e q u e n c y m o d e . F o r e x a m p l e , t h e f o u r t h f r e q u e n c y of t h e 8 degrees of f r e e d o m s i n g l e - e l e m e n t s o l u t i o n , for n = 1, is 0"7894. T h i s f r e q u e n c y a n d its a s s o c i a t e d m o d e are p o o r a p p r o x i m a t i o n s to t h e e x a c t f o u r t h m o d e , i.e. t h e wuv3 m o d e , b u t t h e y a g r e e v e r y well w i t h t h e e x a c t s i x t h m o d e , i.e. v w u l . M a n y similar e x a m p l e s m a y b e f o u n d in T a b l e 1.
F r e e v i b r a t i o n s o f shells o f r e v o l u t i o n using ring finite e l e m e n t s Flat circular plate, clamped
567
at outside edge
T h e l o w e s t six n a t u r a l f r e q u e n c i e s o f a flat circular p l a t e , c l a m p e d a t t h e o u t s i d e edge, for e a c h v a l u e o f n = 0, 1, 2 a n d 3 are g i v e n in T a b l e 2. T h e e x a c t solutions o f t h e e q u a t i o n s o f m o t i o n are g i v e n b y K a l n i n s 8. T h e n o t a t i o n u s e d for defining t h e t y p e o f m o d e is t h e s a m e as t h a t u s e d for t h e c y l i n d e r in T a b l e 1. T A B L E 2.
N O N - D I M E N S I O N A L N A T U R A L F R E Q U E N C I E S OF FLAT CIRCULAR PLATE~ CLAMPED AT O U T S I D E E D G E
= coR(p/E)i;
v = 0.3;
t / R = 0.1
F i n i t e e l e m e n t solutions Single e l e m e n t Mode type
Exact
Multi-element
No. o f t e r m s in p o w e r series 4 6 7 10
16
3
No. o f e l e m e n t s 5
No. o f d e g r e e s o f f r e e d o m 19 10
6
18
22
w w u w v u
0.3091 1.204 2"376 2-696 4.017 4-351
0.3091 1.220 2.376 2.816 4.064 4-353
0.3091 1.204 2.376 2-699 4.032 4.351
0.3091 1.204 2.376 2.699 4.026 4.351
0.3092 1.210 2.434 2.746 4-334 4-684
0.3091 1.204 2.401 2-710 4.122 4.570
0-3091 1-204 2-394 2.703 4-088 4-511
w w
0.6434 1"841 2.052 3-332 3.632 5"250
0.6437 1-889 2.052 3"333 4-709 5-295
0-6434 1.842 2.052 3.332 3.686 5.250
0.6434 1.841 2.052 3.332 3.643 5.250
0.6440 1.858 2.060 3.523 3.980 5-898
0.6435 1.844 2.056 3-405 3.664 5-606
0.6434 1.842 2.055 3.385 3-650 5"513
vu uv
w vu
No. o f degrees o f f r e e d o m
w w vu uv
w vu
w w vu uv
w vu
8
14
17
8
16
20
1-056 2.560 3-195 4-283 4"655 6.142
1.064 2-668 3-235 4-315 6.992* 7-327*
1.055 2-561 3.209 4.294 5.033 6.175
1.055 2.561 3.203 4.290 4.669 6-172
1.059 2.592 3.531 4-889 5.348 7.694
1.056 2-568 3.312 4-509 4.710 7.177
1-056 2.564 3.275 4.438 4-684 6.871
1"548 3"360 4"154 5.266 5"759 7"011
1.554 4-162" 4.473* 5.295 7.498* 8-665*
1.544 3-405 4-154 5.267 6.232 7.036
1.544 3.364 4.154 5-266 5-947 7.014
1.553 3.526 4.428 5.879 6.866 9.111
1.545 3.375 4-250
1.545 3.367 4.222 5.427 5.804 6-745
5.495
5.833 7.944
* N o t m o d e t y p e specified in c o l u m n 2. C o m p a r i n g t h e single e l e m e n t a n d m u l t i - e l e m e n t solutions for a p p r o x i m a t e l y e q u a l n u m b e r s of degrees o f f r e e d o m , it m a y be seen t h a t for t h e t r a n s v e r s e m o d e s , i.e. t h e m o d e s in w h i c h n o r m a l d i s p l a c e m e n t s o n l y are p r e s e n t , t h e r e is little t o choose b e t w e e n t h e t w o e l e m e n t s . F r o m t h i s it is c o n c l u d e d t h a t a n i m p r o v e m e n t in t h e r e p r e s e n t a t i o n o f t h e n o r m a l d i s p l a c e m e n t w i t h a h i g h degree p o l y n o m i a l in t h e s i n g l e - e l e m e n t m e t h o d , r a t h e r t h a n b y piecewise r e p r e s e n t a t i o n w i t h a cubic as is u s e d in t h e m u l t i - e l e m e n t m e t h o d , is offset b y t h e r e d u c t i o n in t h e n u m b e r of degrees o f f r e e d o m a s s o c i a t e d w i t h t h e n o r m a l
568
J.J.
WEBSTER
d i s p l a c e m e n t . T h e s i n g l e - e l e m e n t s o l u t i o n s for t h e i n - p l a n e m o d e s a r e m o r e a c c u r a t e t h a n t h e m u l t i - e l e m e n t s o l u t i o n s b e c a u s e t h e r e is b e t t e r r e p r e s e n t a t i o n of t h e d i s p l a c e m e n t s a n d t h e r e a r e m o r e degrees of f r e e d o m a s s o c i a t e d w i t h t h e i n - p l a n e d i s p l a c e m e n t s i n t h e singleelement method. T h e difference b e t w e e n r e p r e s e n t i n g t h e n o r m a l d i s p l a c e m e n t of t h e whole p l a t e b y a single p o l y n o m i a l or piecewise b y a n u m b e r of cubics, w h e n t h e n u m b e r of degrees of f r e e d o m a s s o c i a t e d w i t h t h i s d i s p l a c e m e n t is t h e s a m e in b o t h cases, m a y b e seen b y c o m p a r i n g t h e f r e q u e n c i e s o f t h e t r a n s v e r s e m o d e s for t h e s i n g l e - e l e m e n t 16 degrees of f r e e d o m s o l u t i o n w i t h t h e 3 - e l e m e n t 10 degrees of f r e e d o m solution. T h e s o l u t i o n s for t h e t r a n s v e r s e m o d e s are i n d e p e n d e n t of t h e r e p r e s e n t a t i o n of t h e i n - p l a n e d i s p l a c e m e n t s b e c a u s e t h e s e d i s p l a c e m e n t s a r e n o t p r e s e n t i n t h e s e m o d e s . T h e n a t u r a l f r e q u e n c i e s for all t h e t r a n s v e r s e m o d e s are p r e d i c t e d m o r e a c c u r a t e l y b y t h e 6 - t e r m p o l y n o m i a l plus t w o rigid b o d y d i s p l a c e m e n t s t h a n b y piecewise r e p r e s e n t a t i o n of t h e n o r m a l d i s p l a c e m e n t b y t h r e e cubics. Spherical
shell
T h r e e s o l u t i o n s for t h e e l e v e n lowest n a t u r a l f r e q u e n c i e s of t h e a x i s y n u n e t r i c m o d e s of a s p h e r i c a l shell, closed a t o n e e n d a n d w i t h t h e edge free or c l a m p e d , a r e g i v e n i n T a b l e 3. T h e e x a c t s o l u t i o n s are g i v e n b y K a l n i n s 9 a n d t h e m u l t i - e l e m e n t r e s u l t s b y N a v a r a t n a 5. T h e s i n g l e - e l e m e n t r e s u l t s were o b t a i n e d b y t h e m e t h o d g i v e n a b o v e . K a l n i n s '9 classificat i o n for t h e b e n d i n g a n d m e m b r a n e or s t r e t c h i n g m o d e s is used. I n t h e s e m o d e s o n l y n o r m a l a n d m e r i d i o n a l d i s p l a c e m e n t s a r e p r e s e n t a n d in t h e t o r s i o n a l m o d e s t h e r e is only tangential motion. TABLE 3. NON-DIMENSIONAL NATURAL FREQUENCIES OF SPHERICAL SHELL v -~ 0"3;
= wR(p/E)t;
Mode No.
Exact (Ref. 9)
t / R = 0"05;
Multielement (Ref. 5)
h a l f - a n g l e --- 60 °
Single element
Modet type
0.931 1.089 1.535 2.360 2.551 2.985 4.023 4.950 5.548* 6.224* 6.862*
B B B B S T B T B S B
1.007 1.391 1.700 2.095 2.386 3.851 4.062* 4.151" 5.962* 6.208* 7.148"
B B B T B B S T B T B
F r e e edge 1 2 3 4 5 6 7 8 9 10 lI
0.931 1.088 1.533 2.348 2.544 -3.497 -4.951 5.230 6.693
0.932 1.094 1.544 2.363 2.548 2.982 3.519 4.971 4.980 5.543 6-734 C l a m p e d edge
1 2 3 4 5 6 7 8 9 10 11
1.006 1.391 --2.375 3.486 3.991 -4.947 --
6.690
1-008 1.395 1.702 2.126 2.387 3.506 3.996 4.159 5.001 6.037 6.729
* N o t m o d e t y p e specified. t B--bending, S--stretching and T--torsional.
Free vibrations of shells of revolution using ring finite elements
569
For the multi-element solution Navaratna represented the shell by 24 curved elements of equal meridional length. The displacement functions are the simplest ones required to maintain continuity between joining elements and the number of degrees of freedom was 98 for the free edge shell and 94 for the clamped edge case. In the single element solution the displacements were represented by four rigid body displacements and three 7-term polynomials. After eliminating the co-ordinates required to satisfy the boundary conditions there are 23 degrees of freedom for the free shell and 19 for the clamped edge shell. Comparing the two finite-element solutions, it may be seen that the single-element solutions are better for the lower frequency modes even though the single-element representation has considerably fewer degrees of freedom than the multi-element representation. CONCLUSIONS T h e results for t h e n a t u r a l frequencies for t h e t h r e e shells g i v e n a b o v e indicate t h a t t h e r e are considerable a d v a n t a g e s in r e p r e s e n t i n g a shell b y a few e l e m e n t s w i t h d i s p l a c e m e n t functions e x t e n d e d b e y o n d t h a t required to satisfy c o n t i n u i t y b e t w e e n e l e m e n t s r a t h e r t h a n using a large n u m b e r of e l e m e n t s w i t h t h e s i m p l e s t d i s p l a c e m e n t functions. F o r a g i v e n n u m b e r o f degrees o f f r e e d o m , usually m o r e a c c u r a t e results are o b t a i n e d b y e x t e n d i n g t h e d i s p l a c e m e n t functions r a t h e r t h a n b y using m o r e e l e m e n t s w i t h simpler d i s p l a c e m e n t functions. Generally, in m o d e s o f v i b r a t i o n in which m o r e t h a n one d i s p l a c e m e n t is present, t h e f o r m o f t h e d i s p l a c e m e n t s is similar, i.e. t h e y h a v e similar n u m b e r s of nodes. F o r this r e a s o n p o l y n o m i a l s o f t h e s a m e degree are used to r e p r e s e n t all t h r e e displacements. I f m o r e t h a n a b o u t s e v e n t e r m s are used in t h e displacem e n t functions t h e e q u a t i o n s m a y b e c o m e ill c o n d i t i o n e d a n d difficult to solve a c c u r a t e l y . U s u a l l y d i s p l a c e m e n t functions of this l e n g t h give a d e q u a t e a c c u r a c y for t h e lower f r e q u e n c y modes. I f t h e higher f r e q u e n c y m o d e s are of i n t e r e s t m u l t i - l e n g t h a r i t h m e t i c could be used to solve t h e equations, b u t it would p r o b a b l y be m o r e c o n v e n i e n t to r e p r e s e n t t h e shell b y m o r e t h a n one element. A n o t h e r r e a s o n for using m o r e t h a n one e l e m e n t is t h a t t h e d i s p l a c e m e n t functions are c o n t i n u o u s w i t h i n t h e e l e m e n t s a n d c a n n o t r e p r e s e n t e x a c t l y discontinuities in d i s p l a c e m e n t or d e r i v a t i v e s of t h e displacements. I t would s e e m to be a d v i s a b l e to divide shells w i t h discontinuities in g e o m e t r y or physical p r o p e r t i e s into a n u m b e r of e l e m e n t s a t t h e points of discontinuity. Acknowledgement--The author gratefully acknowledges the assistance and advice given
by Professor G. B. Warburton during the course of this work. REFERENCES I. P. E. GR~TON and D. R. STRO~m, A.I.A.A. Jl I, 2343 (1963). 2. E. P. PoPov, J. PENZ~N and Z. A. Lu, Jl Eng. Mech. Div., Proc. Am. ,~oc. cir. Engrs 96,
119 (1964). 3. J. H. PERCY, T. H. H. PIAN, S. KL~.IN and D. R. NAV~A~A, A . I . A . A . Jl 3, 2138(1965). 4. R. E. JONES and D. R. STRO~, A . I . A . A . J l 4, 1519 (1966). 5. D. R. N A V ~ A T ~ A , A . I . A . A . Jl 4, 2056 (1966). 6. W. FLi2GGE, Stresses in Shells, p. 317. Springer-Verlag, Berlin (1962). 7. G. B. WAa~BVRTON,J. Mech. Engng Sci. 7, 399 (1965). 8. A. K_~Nr~s, Prec. 4th U.S. hath. Congr. appl. Mech. 1, 225 (1962). 9. A. K ~ i ~ s , J. acoust. See. A m . 36, 74 (1964). 38
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APPENDIX 1
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