Dynamic Analysis of Sandwich Cylindrical Shell
We present dynamic analysis of cylindrical shells and study new mechanical effects in distribution of stresses, strains and displacements. Cylindrical shells are studied in the framework of Kirchhoff-Love hypothesis and the refined Timoshenko model.
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2000
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| Назва видання: | Проблемы прочности |
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| Цитувати: | Dynamic Analysis of Sandwich Cylindrical Shell / K. Cabanska-Placzkiewicz // Проблемы прочности. — 2000. — № 4. — С. 119-127. — Бібліогр.: 7 назв. — англ. |
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irk-123456789-463102013-06-29T15:18:00Z Dynamic Analysis of Sandwich Cylindrical Shell Cabanska-Placzkiewicz, K. Научно-технический раздел We present dynamic analysis of cylindrical shells and study new mechanical effects in distribution of stresses, strains and displacements. Cylindrical shells are studied in the framework of Kirchhoff-Love hypothesis and the refined Timoshenko model. Выполнен динамический анализ цилиндрических оболочек и изучены новые механические эффекты в распределении напряжений, деформаций и перемещений. Цилиндрические оболочки рассматриваются на основе гипотезы Кирхгофа-Лява и уточненной модели Тимошенко. Проведено динамічний аналіз циліндричних оболонок та розглянуто нові механічні ефекти в розподіленні напружень, деформацій і переміщень. Циліндричні оболонки розглядаються на основі гіпотези Кірхгофа-Лява й уточненої моделі Тимошенка. 2000 Article Dynamic Analysis of Sandwich Cylindrical Shell / K. Cabanska-Placzkiewicz // Проблемы прочности. — 2000. — № 4. — С. 119-127. — Бібліогр.: 7 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/46310 539.4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Научно-технический раздел Научно-технический раздел |
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Научно-технический раздел Научно-технический раздел Cabanska-Placzkiewicz, K. Dynamic Analysis of Sandwich Cylindrical Shell Проблемы прочности |
| description |
We present dynamic analysis of cylindrical
shells and study new mechanical effects in
distribution of stresses, strains and
displacements. Cylindrical shells are studied in
the framework of Kirchhoff-Love hypothesis
and the refined Timoshenko model. |
| format |
Article |
| author |
Cabanska-Placzkiewicz, K. |
| author_facet |
Cabanska-Placzkiewicz, K. |
| author_sort |
Cabanska-Placzkiewicz, K. |
| title |
Dynamic Analysis of Sandwich Cylindrical Shell |
| title_short |
Dynamic Analysis of Sandwich Cylindrical Shell |
| title_full |
Dynamic Analysis of Sandwich Cylindrical Shell |
| title_fullStr |
Dynamic Analysis of Sandwich Cylindrical Shell |
| title_full_unstemmed |
Dynamic Analysis of Sandwich Cylindrical Shell |
| title_sort |
dynamic analysis of sandwich cylindrical shell |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| publishDate |
2000 |
| topic_facet |
Научно-технический раздел |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/46310 |
| citation_txt |
Dynamic Analysis of Sandwich Cylindrical Shell / K. Cabanska-Placzkiewicz // Проблемы прочности. — 2000. — № 4. — С. 119-127. — Бібліогр.: 7 назв. — англ. |
| series |
Проблемы прочности |
| work_keys_str_mv |
AT cabanskaplaczkiewiczk dynamicanalysisofsandwichcylindricalshell |
| first_indexed |
2025-07-04T05:32:22Z |
| last_indexed |
2025-07-04T05:32:22Z |
| _version_ |
1836693211141111808 |
| fulltext |
UDC 539.4
Dynamic Analysis of Sandwich Cylindrical Shell
K. Cabanska-Placzkiewicz
Pedagogical University, Department of Mathematics Technology and Natural Science
Institute of Technics, Bydgoszcz, Poland
УДК 539.4
Динамический анализ многослойных цилиндрических оболочек
К. Цабанска-Плашкевич
Педагогический университет, отделение математических технологий и естествен
ных наук Технического института, Быдгощ, Польша
Выполнен динамический анализ цилиндрических оболочек и изучены новые механические
эффекты в распределении напряжений, деформаций и перемещений. Цилиндрические обо
лочки рассматриваются на основе гипотезы Кирхгофа-Лява и уточненной модели Тимо
шенко.
Cylindrical shells with wide range o f geometrical and physicomechanical
parameters are component parts o f m any m odern structures subject to the action
o f different loads. The development o f various vibration models becomes an
urgent problem due to the structural features o f layered shells operating under
different conditions o f m echanical loading. The classical theory o f cylindrical
shells, which is based on the K irchhoff-Love hypothesis, is w idely used for the
evaluation o f the stress-strain state or vibration o f isotropic thin elastic shells.
Among numerous precise models applied to the investigation o f shells made o f
modern m aterials due to their practical validity, visualization, and completeness,
the Tim oshenko shear m odel is used. The behavior o f thick-walled shells under
such conditions is fairly peculiar and complex and requires a more detailed
investigation taking into account the changes in geometry, m aterial behavior,
mode o f loading, conditions o f the boundary, and m any other factors.
The dynamic problem o f elastic homogeneous bodies was presented in [1, 2].
The problem o f simulation o f the acoustic properties o f the larger hum an blood
vessel was considered in [3]. Simple and complex vibration systems were
considered in [4]. The coupled problems o f the therm om echanical behavior o f
viscoelastic bodies under harmonic loading were presented in [5]. The problem o f
nonaxisymmetric deformation o f flexible rotational shells was solved in [6] with
the use o f the classical K irchhoff-Love m odel and im proved Timoshenko model.
Free vibrations o f the elements o f shell constructions were described in [7].
The goal o f this paper is to perform the dynamic analysis o f cylindrical shells
and discover new m echanical effects in the distribution o f stresses, deformations,
and displacements. We present the dynamic analysis o f elastic layered cylindrical
shells o f finite length l for different values o f thickness h. We consider two
models o f deformation o f a straight line shell element that is norm al to the
© K. CABANSKA-PLACZKIEWICZ, 2000
ISSN 0556-171X. Проблемы прочности, 2000, N 4 119
K. Cabanska-Placzkiewicz
undeform ed coordinate surface. The first one is based on the K irchhoff-Love
hypothesis according to which this element remains straight and norm al and its
length does not change in the process o f deformation. The second one is the
improved Timoshenko model, which is also based on the hypothesis o f straight
line, but, in this case, the shell element, initially norm al to the surface, does not
remain norm al to the deformed surface. These kinematic assumptions are
supplemented with static assumptions according to which the norm al stresses on
the squares that are parallel to the coordinate surface, as compared with the
stresses on the other squares, can be neglected. The inertial forces associated with
the displacement o f a surface element along the coordinate axes are also taken
into account. The layers o f the shell are deformed without m utual separation on
the entire surface o f contact.
Due to the assumptions made, the displacements and deformations o f
arbitrary points o f the shell are determined by the displacements and deformations
o f the coordinate surface. Thus, the displacements o f the shell points located at a
distance z from the coordinate surface are determined as a near function o f the
displacements o f this surface:
U ( s , 6 , r , t) = u( s , 6 , t) + r p i ( s , 6 , t),
V ( s, 6 , r , t ) = V ( s , 6 , t ) + rp 2 ( s, 6 , t ),
W ( s , 6 , r , t ) = w( s , 6 , t ),
where p i , ip 2 are the angles o f rotation o f the norm al in the planes 6 = const and
s = const, respectively, for the classical version o f the theory, or the full angles o f
rotation for the im proved version o f the theory with shear deformations taken into
account.
Investigations were carried out for both models for radial displacements w in
the cylindrical coordinates r , z , 6 (—h / 2 < r < h / 2 , 0 < z < 1,0 < 6 < 2n ) under
axisymmetric loading.
O ne-L ayered C y lindrical Shell. According to the assumptions o f the
classical K irchhoff-Love theory, we have e r = e zr = 86r = 0, the norm al stresses
o r = 0, and w (z , r ,6 , t) = f ( z , r , t). In the m athematical m odel o f the problem on
the basis o f the classical K irchhoff-Love model, the system o f conjugate partial
differential equations describing the phenom enon o f small transverse vibrations in
the considered physical model has the form
Eh d u
1
D
v 2 d z 2
d 4 w
v d w , d u
------ ;— P h — t
R d z d t
= 0 ,
Eh
dz4 ( i - v 2 ) R \ R
w du
— + v — l + p h
d z
d 2 w
d t2
(1)
= 0,
where
D =
E h 3
12(1- v 2 ) '
120 ISSN 0556-171X. npodneMbi npouuocmu, 2000, № 4
Dynamic analysis o f sandwich cylindrical shell
A solution o f this problem was obtained in the following form:
— n t ( n n z \? 'In1 ------ (
n=l
dw
dz
(2)
Function o f stresses has been accepted as
T N
O
O
O,
X'' X '' /- \ • n n z .= 2 2 O r ( r ) s in ----- sin rnt,
t=0 n=l 1
T N
= 2 2 o z ( r )
t=0 n=l
T N
= 2 2 o o ( r )
n n z .
cos----- sin rnt,
I (3)
t=0 n=l
n n z .
cos----- sin rnt,
I
where o r ( r ), o z ( r ), o e ( r ) are functions varying across the thickness h and
length l o f the shell at time t.
U nder the assumptions o f the im proved Timoshenko model, in which the
influences o f forces o f rotational inertia and shearing deformation are taken into
account, one has deformations e zr ^ 0, e 9r ^ 0 and norm al stresses o r = 0. The
m athematical m odel o f the problem on the basis o f the im proved Timoshenko
m odel is represented by the following system o f conjugate partial differential
equations
D d 2 ÿ z . . ,r , J d w I p h 3 d 2 ÿ z = 0
D — T + I c ^ — — ÿ z j - — - d T = 0-
p h
dz
d 2 w
d t2
l
k ' Gh
d 2 w
d z 2
# z
dz
+
d t
Eh
( l - v 2 )R
w du''.
— + v — | = 0.
R dz
(4)
In this case, we found a solution o f the problem in the form
w = 2 e"
n=l
ÿ z = 2 ‘
n=l
^ -- 2ÿ z p h 3 d 2ÿ z
D — h r - + — ------- ^ + k ' G h ÿ ,
dz l 2 d t2
I G h
Eh
cos( « nt + p n )
d 2 w d 2 wdu \
2 2 I w + v —— | + p h 2 2
L(l — v 2 )R 2 \ d z j d t2 d z2
(5)
cos( rn n t + p n ).
Here, w , u are displacements o f the shell, p z is the angle o f rotation o f a cross
section o f the shell, p is the mass density o f the m aterial o f the shell, h is the
l
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K. Cabanska-Placzkiewicz
height o f the shell, v is Poisson’s ratio, G is the K irchhoff modulus o f the
material o f the shell, E is Y oung’s modulus o f the material o f the shell, k ' is the
shear coefficient, and t is time.
First, we consider a cylindrical shell under load q n = q 0 sin(n n z / 1)
distributed symmetrically with respect to z-axis. Elastic cylindrical shell with
finite length l , rigidity D , and external radius R is considered with free supports
on its ends:
w| 7=0 = 0 W 7=1 = 0 (6)
Num erical calculations are perform ed for the following data: l = 120 mm,
h = 2; 6 mm, E = 2.1-1011 N /m m 2, v = 0.3, n = 1 ,2 , 20.
Some results concerning the solution o f the problem are shown in Figs. 1 and
2. The diagrams present free vibrations for displacements w(z , t ) in the middle
section o f the cylindrical shell for different thicknesses h / R = 1/10; 1/30 and the
initial condition w 0 = 0.02sin(n n z / 1) [m]. The two-dimensional diagrams
display the distribution with length 0 < 7 < 0.2 l at time t = 0.48 s for various
n , d = l / nh. The diagrams m arked with “a” correspond to the cases where the
classical K irchhoff-Love model was applied, whereas the diagrams marked with
“b” correspond to the cases where the improved Timoshenko model was used.
w b
Fig. 1. Distribution of free vibrations of the cylindrical shell for h / R =1/10.
w
0 .0 2
b
6=60
0 .0 1 5 6=6
6=3
0 .0 1
a
8=60
0 .0 0 5
6=3 6=6
5 10 15 20 z ,
122
Fig. 2. Distribution of free vibrations of the cylindrical shell for h / R =1/30.
ISSN 0556-171X. Проблемы прочности, 2000, № 4
Dynamic analysis o f sandwich cylindrical shell
Analyzing the results for h / R = 1/10, 6 = 1 , 2 (Fig. 1) and h / R = 1/30,
6 = 3, 6 (Fig. 2), one can note that, as n = 1, 10, 20 increases, the free vibrations
w decrease and the difference between the results obtained for the Timoshenko
and K irchhoff-Love models increases. As 6 decreases, the difference between
the values o f displacement for the two models considered increases.
The distributions o f stresses a z , a r , a 0 , t rz and deformations e r , e z for
cylindrical shells for z = 0.51 across the thickness — h / 2 < r < h / 2 are shown
in Fig. 3. The spatial curves o f stresses and deformations are presented for the
external load q n = q 0 sin(n n z / l ), which affects the cylindrical shell, with respect
to time. The components o f stresses and deformations vary nonlinearly across the
thickness. The tangential stresses t rz vary across the thickness according to the
parabolic law. The distribution o f stresses t rz is symmetrical w ith respect to the
z = 0.51 axis. The normal stresses depend on r. Strain distribution e z , e r (both
axial and radial ones) also have the form o f a centrally symmetrical parabola.
Considering the section z = 0 .2 1, we see that the longitudinal a z , radial a r , and
circular a g stresses reach values that are smaller than those for z = 0.51 by
approximately 60%, 6%, and 54%, respectively. The tangential stresses t rz for
z = 0.21 reach values approximately 5% smaller than for z = 0.51. For z = 0.21,
the deformations along the radial direction e r reach values approximately 75%
smaller than for z = 0.51. For z = 0.21, the deformation along the axial direction
e z reaches a value approximately 50% smaller than for z = 0.51.
Fig. 3. Distribution of stresses a z, a r , ae, t rz and deformations er , e z for cylindrical shells for
z = 0.5 l across the thickness — h / 2 < r < h / 2.
ISSN 0556-171X. npoôneMU nponnocmu, 2000, № 4 123
K. Cabanska-Placzkiewicz
The distributions o f the radial displacement w( r , t ) across the thickness
h , z = 0.51 at time t are shown in Fig. 4. The diagrams m arked with “a”
represent the distribution according to the Timoshenko model, and the diagrams
m arked with “b” describe the distribution according to the K irchhoff-Love
model. In the case o f the Timoshenko model, the radial displacements vary across
the thickness and decrease with time. In the case o f the K irchhoff-Love model,
the radial displacements are described by a constant function across the thickness
and also decrease with time.
Fig. 4. Distribution of the radial displacement w(r, t) across the thickness h, z = 0.51 at time t.
For cylindrical shells with h / R > 1/10 and d < 3 subjected to nonuniform
loading, it is necessary to use the Timoshenko model. The application o f the
classical K irchhoff-Love m odel in these cases m ay lead to significant errors.
Sandw ich C y lind rical Shell. In the case o f a system o f two cylindrical
shells coupled by a viscoelastic interlayer, the m athematical model o f the problem
corresponding to the Timoshenko m odel is represented by the following system o f
coupled partial differential equations describing small transverse vibrations o f the
system:
w( r , t ) / 1 0 3 q ohE 1 w( r , t ) / 1 0 3q 0hE 1
a b
(7)
p 2 h2 + ------ ^ --------- - + V
d t2 k 'G 2h2 [ d z 2 d z J (1 — v 2 )R2 VR2 d z
d
124 ISSN 0556-171X. npo6neMbi npouHocmu, 2000, № 4
Dynamic analysis o f sandwich cylindrical shell
where R i is the radius o f the internal shell I, R 2 is the radius o f the interlayer,
R 3 is the radius o f the external shell II, hi and ^2 are the heights o f shells I and
II, P i and p 2 are the mass densities o f the m aterials o f shells I and II,
w 1n, w 2n , u 1n ,an d u 2n are the displacements o f shells I and II, ^ in and ^ 2n are
the angles o f rotation o f cross sections o f shells I and II, E i and E 2 are Y oung’s
m oduli o f the materials o f shells I and II, k ' is the shear coefficient, k is the
coefficient o f elasticity o f the interlayer, and c is the coefficient o f viscosity o f
the interlayer.
The free vibration o f a system o f two cylindrical shells coupled by a
viscoelastic interlayer was determined with the use o f the Timoshenko model as
follows:
X
w 1n = 2 e ~ tlnt lW1n II ̂ n |[cos(O nt + P n + X in X
n=1
X
u 1n = 2 e ~Vnt IU in 11 ̂ n |[cos(O nt + P n + $ in X
n=1
X
- n^ 1» = 2 e Vnt I H Ф n |[cosO nt + ^ n + e in ),
n=1
X
w 2n = 2 e ~ tlnt |W2n 11 Ф n |[cos( m nt + ^ n + X 2n X
n=1
X
u 2n = 2 e ~ ylJ |U 2 n 11Ф n |[cos( m nt + ^ n + # 2 n X
n=1
X
2n = 2 e ~ nnt | XV2n || Ф n |[cos( Ю nt + ^ n + # 2n X
n=1
where
I
=nФ 2
1 V
h h2
12R1
+
(8)
+ c(W1n - W2n X
I“1 l I h 2
dx I J W 12nw 1 0 + U in u 1 0 + 1 2 ^10 +
J 0 1 12n212R1
+ W2nw 20 + U 2nu 20 + TT"2" ̂W2n^ 20 + c(W 1n “ W2n )(w10 “ w 20 )
12R 2
d x ,
X 1n = arg ^ 1n > X 2n = arg ^ 2n > # 1n = arg ^ 1n > # 2n = ar§ ^ 2n =
0 1n = ar§ ̂ 1n > 0 2n = ar§ ̂ 2n > (P n = ar§ Ф n >
and v n = г n n ± ^ n are the complex natural frequencies.
ISSN 0556-171X. Проблемы прочности, 2000, № 4 125
2
2
K. Cabanska-Placzkiewicz
t , s
t , s
w, m
O .I
0 .0 7 5 \ Wi
0 .0 5
0 .0 2 5
w 2
- 0 . 0 2 5
A 0 . 0p t ~ ■ t^ T ^ 2 y 0 .0 0 3 " 0T 004 ’
OO1
c
w, m
o . i
0.075
0.05
0.025
-0.025
-0 .05
Fig. 5. Free vibration of a system of two cylindrical shells coupled by a viscoelastic interlayer for
z = 0.051 and different thicknesses of internal layers: a) R1 = 0.02 m, R2 = 0.03 m, R3 = 0.04 m, and
h = 0.01 m; b) R: = 0.02 m, R2 = 0.04 m, R3 = 0.05 m, and h = 0.02 m; c) R1 = 0.02 m, R2 = 0.05 m,
R3 = 0.06 m, and h = 0.03 m; d) R1 = 0.02 m, R2 = 0.06 m, R3 = 0.07 m, and h = 0.04 m.
126 ISSN 0556-171X. npo5neMbi npounocmu, 2000, № 4
Dynamic analysis o f sandwich cylindrical shell
According to the K irchhoff-Love model, the free vibration is described by
(2).
Some results concerning the time dependence o f the distributions o f
displacements w and W2 for the external layers I and II in the system o f two
cylindrical shells coupled together by a viscoelastic interlayer for different
thicknesses o f the internal layers in the case z = 0 .51 are presented in Fig. 5. The
thicknesses o f the external layers do not change. The internal layers are m ade o f
8 2the soft material w ith E = 10 N/m m , v = 0.3 and are described by the
V oigt-Kelvin model. The external layers o f the cylindrical shell are made o f an
11 2elastic material with E i = E 2 = 2.1-10 N/m m , and v = 0.3. The displacements
o f the internal layer for the thickness h = 0,01 m are shown in a Fig. 5a and reach
values approximately 15% sm aller than those in Fig. 5b for the thickness
h = 0,02 m. The displacements in Fig. 5b reach values approximately 8.5%
smaller than those in Fig. 5c for h = 0,03 m. The displacements in Fig. 5c reach
values approximately 5.3% smaller than those in Fig. 5d for h = 0,04 m.
In conclusion, it m ay be noted that, as the thickness R 3 o f the interlayer
increases, the free vibration o f a sandwich system decays m ore slowly w ith time t.
Р е з ю м е
Проведено динамічний аналіз циліндричних оболонок та розглянуто нові
механічні ефекти в розподіленні напружень, деформацій і переміщень.
Циліндричні оболонки розглядаються на основі гіпотези К ірхгоф а-Л ява й
уточненої моделі Тимошенка.
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упругих телах. - Киев: Наук. думка, 1981. - 284 с.
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конечных размеров. - Киев: Наук. думка, 1978. - 264 с.
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Hydromechanics, National Academ y o f Sciences o f Ukraine).
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1971. - 558 с.
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Наук. думка, 1981. - 288 с.
6 . P a n k r a to v a N. D ., N ik o la e v B ., S w ito n c s k i E . N onax isym m etrica l
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Received 02. 10. 99
ISSN 0556-171X. Проблемы прочности, 2000, № 4 127
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