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Static and transient response of cylindrical shells
Thin-Walled Structures 5 (1987) 157-179
Static and Transient Response of Cylindrical Shells
A. Bhimaraddi* Department of Civil Engineering, University of Melbourne. Parkville. Victoria 3052~ Australia (Received 20 April 1985; revised version accepted 27 August 1986)
A BS TRA CT Static and transient analysis ofcomposite ~Tlindrical shells" is presented using a recently proposed shear deformation theory. The dynamic response is obtained by employing the numerical time integration scheme due to Newmark. The results obtained by using classical shell theory ( ( S T ) and Mindlin-type shear defbrmation theory (SDT) are compared with those obtained by using the proposed theory. The comparison studies reveal that the linear stress distribution, as assumed in CSTand SDT, differs considerably from the predicted nonlinear distributhm qC'theproposed theory.
NOTATION a,L -~-~ Ex, Eo, Ez G xo, G x~, G ~z h
hl, h2, h3 ITl, Fl
Mean radius and length of cylinder. Localized load dimensions (see Fig. 1). Young's moduli. Shear moduli. Total thickness of shell. Thicknesses of the individual layers. Axial (half) and circumferential wave numbers.
* Present address: Department of Civil Engineering, University of Canterbury. Christchurch l, New Zealand. 157
Thin-Walled Structures 0263-8231/87/$03.50 © Elsevier Applied Science Publishers Ltd, England, 1987. Printed in Great Britain
158
.4. Bhimaraddi
N~,No, N~o, No~ } M~, Mo, Mxo, Mo~ Stress-resultants. M '~, M 'o, M 'xo,M 'o~ Applied loads on the outer and the inner surface ot the qzo, qz~ cylinder in the z-direction~ t Time coordinate. Pulse duration time. to tp Fundamental natural period (= 2n-/~)o LI, V, W Displacement components in the x-, O- and z-directi~ m s Generalized displacement quantities. U0, /~1. ]20, Vl, W0 x,O,z Cylindrical coordinate system. Relative thickness ratios in laminated cylinders [3, = h~_/h,,[33 = h2/h3,[3 = [3~ :----/3:,. Shear strains in cylindrical coordinates. ]/xo, 3/xz, "Yoz 6~ Percentage error in classical shell theory. Percentage error in shear deformation theor}.. At Time step-size used in numerical time integration. Normal strains in the x-. 0- and z-directions. Ex, ~-o~ Ez Poisson's ratio. I~ ~o Angular frequency. P Mass density of the material. Mass densities of individual layers. pl, p2, p3 Normal stresses in the x-, 0- and z-directions. O'x, C1"O~O" z Shear stresses in cylindrical coordinates. "grO~ T~z, TOz
( To, T, AT, Tp) =:
(10, t, ~,1, t,,)
L x
(a,F,~) = - - ( u , v , w ) aqo l
q0
1 INTRODUCTION The increased use of composites for high performance design applicanons necessitates more realistic prediction of the response characteristics of composite structures. It is possible to treat composite shells constructed of laminates by applying three-dimensional elasto-dynamics to each individual
Static and transient response of cylindrical shells
159
layer in conjunction with appropriate continuity and boundary conditions. However, it is well known that the complexities involved in the mathematical treatment of laminated shells using three-dimensional elasticity make it impractical to carry out such an analysis. To avoid this complexity, it is the usual engineering practice to make certain assumptions regarding the kinematics of deformation in the thickness direction of shell. For example, in the classical shell theories (CST)I-9 it is assumed that normals to the midplane before deformation remain straight and normal to the midplane after deformation; the implication is that the thickness shear deformations are negligible. Recent developments in the analysis of plates and shells laminated of fiber-reinforced materials indicate that thickness has more pronounced effects on the behaviour of composite plates and shells than on isotropic laminates. Also, due to low transverse shear moduli relative to the in-plane Young's moduli, thickness shear deformations are even more pronounced in composite laminates. Hence, a mathematical model to analyse composite laminated shells, in order to be closer to reality, must incorporate the effects of transverse shear deformation. There exist many shell theories that account for transverse shear strain effects.~°-~5 These shear deformation theories (SDT) are based on the assumption that the normals to the midplane before deformation remain straight but no longer remain normal to the midplane after deformation; the implication is that the in-plane displacement components are assumed to be linear functions of the normal coordinate (z) and warping of cross-sections is not allowed. For accurate determination of response characteristics of shells, in cases such as when accurate higher order frequencies are required 16or when there exists an appreciable thermal gradient through the thickness, ~7the above mentioned shear deformation theories are found to be inadequate. Thus, there have appeared many higher order theories ~6-27which account tor transverse deformations and warping of cross-sections. Usually, higher order theories are based on the assumed form for displacements--taken as power series expressions in z. It may be stated here that most of the available higher order theories employ too many unknown displacement parameters in order to achieve parabolic variation for transverse shear strains. In addition these theories, with the exception of ref. 17, do not satisfy shearfree conditions on the outer and the inner surfaces of the shell. Further discussions on the applicability and the accuracy of such formulations for plates and shells can be found in ref. 25. The higher order plate theories in which in-plane displacement components are assumed as cubic polynomials in z, resulting in parabolic variation for transverse shear strain distribution, can be found in refs 28-31. These theories employ fewer unknown displacement parameters, equal to
160
A. Bhimaraddi
those used in shear deformation theories, 11.15and they also satisfy shear-free surface conditions. However, it is to be noted here that, owing to the difference in the assumed in-plane displacement forms, classical plate theory equations can be obtained as a special case from ref. 29 but not from refs 30 and 31. Also, the displacement forms chosen in ref. 29 are more versatile than those of refs 30 and 3 l, in a sense that almost any {:(z) (see eqn (21 below) whose first derivative vanishes at +-h/2, and is non-zero elsewhere. can be used in ref. 29, resulting in a corresponding higher order theory. Such various forms for sO(z) and the procedure for systematic reduction ot equilibrium equations in terms of stress-resultants for plates and shells using three-dimensional stress equilibrium equations have been given in ref. 32, A higher order theory for cylindrical shells, similar to that for plates. > has been given by the author 33'34in the context of free vibration analysis. The superiority of the proposed shell theory (PST) has been demonstrated in re[. 33, by comparing the results obtained by PST, SDT and CST with those obtained by using three-dimensional theory of elasticity, 3s in the context ot free vibration analysis of isotropic cylinders. A similar comparison lk~r orthotropic homogeneous and laminated shells has been given in ref. 34. In this paper the complete formulation of the theory, for generally orthotropic shells and also further comparison of PST, SDT and CST in the context ol static and forced vibration analysis has been dealt with. A solution to the forced vibration response has been obtained by numerical integration of the resulting system of dynamic equilibrium equations using Newmark~ method. 36,37
2 G O V E R N I N G E Q U A T I O N S OF MOTION The components of displacements are assumed as Mllows (refer to the Notation) u (x, 0, z, t) = u0(x, 0, t) + {:ul(x, 0, t) - ,'.w0, ~ a-Z~ Wl,, 0 w ( x , O, Z, t) = Wo(X, O, t)
where : z
( 4z2/ l-3-
~tl
Static and transient response o f cylindrical shells
16 1
In these equations a comma denotes differentiation with respect to the letter(s) followed by it. By writing out the expressions for strains 32'3339using the above displacement components one can easily verify that the transverse shear strains, and hence stresses, vanish on the outer and inner surfaces of the cylinder and that they vary parabolically across the thickness of the shell. This obviates the use of shear correction factors, usually employed in Mirsky-Herrmann type" shear deformation theories. It may be said here that the form of in-plane displacement components in eqns (1) is not the only choice. There could be other possible alternatives to choose for ((z) to accomplish the above mentioned purpose. The other forms for ((z) may be found in ref. 32. Also, if one wants to have a greater number of terms in the w-expression, corresponding terms should be added to the u- and v-expressions to have the shear-free surface boundary conditions satisfied. This will, of course, result in a more accurate but more complicated theory involving the consideration of thickness normal strain effects, and a greater number of generalized displacements. Hence, to limit the complexity of the problem to a reasonable degree, the w-expression has been chosen to be a constant across the thickness of shell. The constitutive relations corresponding to a monoclinic type of material (or for any layer in the case of a laminate) are of the form, (r~ = (_'lte.~+ Cl2Eo+ C~ey~o ; cro = CI2E~+ C22~e+ C26y.o
(3)
"F~0 = CI6Ex+
C26Eo + C66")lxo ; rxz = C44Txz + C45Toz ; for = C45")/xz+ C55Toz
We have the following definitions for stress-resultants appropriate to the present method of analysis 32'343~ =
(or,, o-0) .___5
M i~ M'o
Nxo M.'~o
= f
It'] o+Z/2 --7]
(l'sC)rx°dz
(4)
a Q . = aMx, x + (M.o + M~x), o + PsUo, t, + p llu 1, . - - p6Wo ....
(5) a2Qo = aMo. o+ aZ(Mxo+ M o . ) , ~ + (apS + p 6 ) v o , . + (ap ~) + p~2) v l , . -
p6wo. o.
162
,4. Bhimaraddi
and the equilibrium equations in terms of stress-resultants can be writter~ as,
32
aN ..... + No~. o = - a y
pu, u ( - - /+s z
dz
aN~o, .~+ No. o = - a I
pv, t, (--~-[--)a+ z
-dz
aM', ~+M'ox, o - a
aM'o,~+M'o,,--a
~'xz~l a * c ,,
.r
j
roz~l --~-Z
1
aM ..... + ( M ~ o + M ~ ) , x o + - M ~,oo+ a p(azu, ,.,,+ zv. o, +
,,) (
I-<.-)
dz ...... a
[u,,,~'i~"\d: t
dz = -a
f
y
Ov,,,~\
+~-- dz
Cr0dz .... q
dz
where ~ = ~,= =
4.,,: 1-~-)
h ; q = a(q:,, - %,I t - ~ ( q : o . q : , )
H e r e , q zoand q z~are the applied loads in the z-direction on the outer and in nc r surfaces of the shell, respectively. Substitution of stress-resultants (41. expressed in terms of the generalized displacements of the problem, in eqn~ (6) results in the following set of dynamic equilibrium equations~ L8
=
P
where 8 + = (Uo, Vo, U . V . W o ) pT = (O,O,O,O,q)
and the operators L u are symmetric and have the following forms:
163
‘L,
.I L13
L.32
=
L2.1
a2c:
il = .J1 J u2cg
-$().
+
c7422
Ml +
c:;
‘I
1
I?
L,4
f( 1, ,x+
L 23
Li.4
.
+
c:,
Cih
0
0
1
J( 1.HH-
c-i_,
0
c-4126
_
c$ + cg.j [ cy,+ cz
I
C::
A(19
XH”
1(}4
A. Bhimaraddi
l
[]
a { ).o~o+
'~:
( }.,-'
(";2 ]
( >
/
a-
L~s : -~-'?(aeC'~:() ..... +
-,~
Ii1
_
J
( )-o-
I r,l~,
[1
L
'
L " 2~,
P ['
2aC~{ ), ,,,o + (2a("}2 r {-3,~
~-2
"aC '~
a~'('~]{
}
+a2p~( ) .... ,+P6( ).oo,,_ a 2 p4 { )..)
L45 ]
C iB
I
c.+
~'~ 26
llii1
a-
c~2
1
4-
g 44
I a-Cil v ,251
{~,;,
'
~{ ).,, . . . . . .
L c L-J
[ L~3]
~g
'
.-:{ }.,,,~
, -'? [a(
t
'46,1"1
a - ( ;g 1 -:( i
2
L c~ J
tc~J
a"(44
1
I ~{
a2{ ) +
L c~
),,,
o :1
w h e r e . ( ) ..... etc., r e f e r to derivatives o f the c o l u m n matrix o f the g e n e r a l i z e d d i s p l a c e m e n t s s h o w n in e q n (7) a n d , cikl = IkCij ; pk __ Ikp
:~)
Staticand transientresponseof cylindricalshells
165
where Ik are the various integrals to be evaluated between the limits - h / 2 to + h / 2 , as defined in the Appendix. The boundary conditions along the edge of the shell require that either one member of each of the following six pairs, or six linearly independent combinations of them, must be specified, Along x = constant: N,u() ; Nxovo ; Mi~ui ; Mi~oVl ; Qxwo ; Mxwo, x Along 0 = constant: Nod~() ; N~vo ; M'oxul ; M',v, ; Qowo ; Mowo, o
The governing equations of the classical shell theory can be obtained by deleting the third and fourth rows and columns, which correspond to u~ and v~ displacdlnent parameters, from the matrices of eqn (7). Similarly, all o t h e r relations, such as stress-resultants, boundary, conditions, etc., can be obtained by ignoring the terms containing u~ and v ~.
3 SOLUT1ON OF T H E E Q U I L I B R I U M E Q U A T I O N S The closed-form solutions to the equilibrium eqns, (7), with generally orthotropic (monoclinic) material are difficult to obtain. However, specially orthotropic, simply supported shells render possible the closed-form solution in the form of a double Fourier series, each term of which is explicitly defined by a set of simultaneous equations. The shell is assumed to be simply supported at x = 0 and x = L and the solution is assumed to be ,,,)(x.o,t)=
T. n
.,(x.o,t)
=
Vo(X,o,t) rn
-T.
v,,,..(t),b..,,,(x.o) n
E Olmn(t)(l)lrnn(x,O); Vl(X,O,l)~ ~ n
w0(x. o, t) :
m
n
m
~ Vlmn(t)(Ik2mn(x,O) rn
(9)
E W°,nn(t)OP3mn(X, O) n
m
where (~) Iron
= cos(mTr.r/L)cos(nO)
" ~2,,~ = sin(rnrrx/L)sin(nO)
qb3m, = sin ( m r r x / L ) cos (nO) The spatial variation of the loading has been assumed in the form of a double Fourier series to be q(x,O,t) =
~
~
n
m
qm,(t)sin(mrrx/L)cos(nO)
(10)
A. Bhimaraddi
166
where the terms q~), in the above series can be determined as qm=(t) = 0
q,~(t)
= ~
, m = n =0
1 y'"
qm,,(t) = ~ -
q(x.O,t)dO
IJ" .
m :~: (I. n ..... i)
q(x,O,t)sm(mrrx/L)cos(nO)d.vdO
~
~
., ,~ 4= !~ t~. j ,~
Substitution of eqns (9) and (10) in eqns (7) results in the following system of second order ordinary differential equations, variable in time, for each se~ of ( m , n ) values, MA+KA- q
~!!)
where K and M are the symmetric stiffness and mass matrices of order 5 and we have the following definitions for A. A and q,
A =
"
Uornn, V(Imn, UI .... Vlmn" W,)rnn l h
q
=
tO,O,O,O,(a+~)q,,~(t)}
Note that the superposed dots indicate differentiation with respect to troy.:. For a given set of (rn, n) values the A vector can be obtained at vari~ms desired values of time by solving eqns (12) using the time integration scheme due to Newmark. 36.37By summing the series over all (m, n) values up to the desired degree of accuracy, displacement and stress distribution in space and time can be obtained for a specified variation of the external loading m space and time. However, for a static problem the A vector is computed by solving the set of algebraic equations which are obtained from eqns (12) by ignoring the time dependence of displacements and loads. Lastly, it should be said here that the transverse shear stress distribution across the thickness, in the case of SDT and CPT, has been obtained by lhst computing the values of shear forces and then distributing them according to the parabolic law. However, the actual shear stress distribution is constant according to SDT and is zero according to CPT. The spatial location oi the
Static and transient response of cylindrical shells
167
points to which the values of various displacements and stresses (referred to in the discussion of numerical results) correspond, is as given below,
( w , ~ . g / o )
at
at
x L
-
z h
-
0 7r
x L
at
x
I 2'
-
-
0
0
,
=
0
z h
-
z h
-
0 rr
-
1 2
z h
-
0 7r
-
0
L
~z
at
x L
-
to..
at
x L
-
0 rr
1 2'
-
x ~o at ~ - = 0
0 n-
1 2
-
0
: h z
'
-
1 -
h
2
4 DISCUSSION OF NUMERICAL RESULTS In all of the numerical results presented herein, zero initial conditions fl)r displacements and velocities have been assumed. The numerical results for transient response have been obtained by using a time step-size (AT) equal to 1% of the predominant natural period of the system. The accuracy and the correctness of the algebra and the computer coding have been ensured by comparing the results with those available in the literature. 9 The S D T results in the present study have been obtained by using a shear factor value of ~.2/12. There are different shear factor values available in the literature for homogeneous and laminated plates and shells which have often been obtained for a particular problem under consideration. Hence, it b e c o m e s extremely difficult to choose these values for the problems dealt with in this paper. Usually, a value of 7r2/12 is used in the absence of a rational explanation as to the use of other values than 7r2/12. Further discussion on the use of shear factor values may be found in ref. 38. Detailed studies on the static and dynamic response of cylinders with various geometric and material parameters were undertaken. 39 However,
168
A. Bhimaraddi
only a few results have been given here for the sake of brevity. The influence of material orthotropy has been considered using E x / E o values of 1.25 and 40 while keeping G~o = Gx~ - G ~ : '._,Eo, tz~, = '. p~ .... P2 : P~ :- P. 4.1 Static r e s p o n s e o f a shell subjected to localized loading
For a localized static load of intensity q0 applied over an area a x {~ with tile centre at x = x0 and 0 = 0, as shown in Fig. 1, q~. is given by q.,. = 0
. m = n = 0
a + - ~h/ '~ qm.
qmn
_
=
2bq0 mTr2Sin(mTr-a9 / - L ) :si n ( rn7rx o/L). tn 4= q}. n - - ( )
8q0 s i n ( m T r ~ / 2 L ) s i n ( m r c x o / L ) s i n
7.g2mn
[ ,,
•
m ~ (). t~ i
~i
In the numerical results presented here a centrally applied localized load of square shape (~ = b; xo = L / 2 ) has been considered. Figures 2 and 3 show the influence of h / a ratio on radial displacement ( ~ ) and axial stress ( ~ j . The percentage differences (6c in CST and 6. in SDT) have t~cen c o m p u t e d as follows, 6c=
value from CST ) value fromPST - 1 ×100
6~:
value from SDT ) value fromPST - 1 × l(~l
-~ b ~
"~-x°--W axis
X
L
Fig. 1. Description of the localized static loading.
Static and transient response of cylindrical shells
- - 6 s
Ex /EB= 25
.... Sc Ex/Ee = 25
L l e = 1.0 8
O=l.O b
-0.6
I
6
4 -0.4
m o g e L . . . . _ _
~
¢,
-20
\oj,.-
/
// /I
2
0/90 ..
~%~/ (b'/~(3 .
_ b _ 1 ~./~p) ' / ~--~--~ .%, / e°,; " ,~I II 0\/
-30
x:lo
/
\ ~I I
Lie -- !.0
-40
-0.5 ,3
169
/I
/~+,/~,J
~.
-10
-0.3
0 ,
0.05
0.10
O.05
0.15
O.10
O.15
h/a
h/~
Fig. 2. Variation of radial (~) displacement and the corresponding percentage differences in SDT ((5~)and CST (8~) for different thickness-to-radius (h/a) ratios of cylindrical shells under localized static loading (Homoge. = homogeneous).
---
~(I)~10
E,/E0o 25 L/~
-0,2C
: 1,0
/~/
/3~c~y//.
p:l.O
P -40
-010
-012
x"/ / / /
///
,S s
------
Sc
Ex / E 8 : 25 L/a -- 1.0
13:1.0
~_i_i ~-~-~
~,+~
-30
.o~"~ F / " .i j~×
- -
..-//+o¢ ~o\~
ILO
/o\"
-20
-0.08
//'./
• /
//// -0.0~
V, o!
-10
,'/..S..;
/
>X~U'_~
~-
/ I
I
}
0,05
0.10
0.15
h/a
0.05
0.10
0.15
h/o
Fig. 3. Variation of normal (~x) stress and the corresponding percentage differences in SDT ( S j and CST (8c) for different thickness-to-radius (h/a) ratios of cylindrical shells under localized static loading.
,4. Bhirnaraddi
170
Also, due to its superior performance (to CST and SDT) in the case ot h o m o g e n e o u s isotropic shells when compared with elasticity results, 3~'~ ~s and due to the nonavailability of elasticity results for composite shells, comparison of various shell theories has been made taking PST as vhe standard solution. It may be seen from Fig. 2 that the ~ response for homogeneous shetis is high c o m p a r e d to that of layered shells. Also, if: values for three-layL:red shells are less than those for two-layered shells, except for h/a < 0"05. t'ot which the 0/90/0 shell has higher ~ values than two-layered shells. Further. we observe that CST underestimates (8~.<:0) ~ values, whereas S D I overestimates (8~>0) these values. The 8~ and 8~ values increase with increasing h/a. The maximum value of 8~ is about - 5 0 % and that ol #5, is about + 7 % as may be seen from Fig. 2. These values will be still higher h~l higher values of E~/Eo and for more concentrated loadings. From Fig. 3(left) we note that the stress response is maximum Io~ h o m o g e n e o u s shells as compared to layered shells. From Fig. 3(right) we note that both CST and SDT underestimate axial stress value when c o m p a r e d with PST. The errors increase with increasing h/a. with a m a x i m u m value of - 3 5 % for h/a = I). 15. These errors will be still higher for higher h/a and E,/Eo values and the more concentrated the load becom,=s. It is evident from Figs 4 and 5 that as the thickness of shell increases the normal stress and the in-plane displacement variations across the thicki~csof shell b e c o m e more and more nonlinear. It has also been observed thal t~)[ higher E,/Eo ratios, for more concentrated localized loads, and for small L/a values, stress and displacement distribution becomes more and m,~,rc' nonlinear. Hence, for such parameters the predictions based on the lfi,cm distribution, as assumed in CST and SDT, differ considerably.
4.2 Dynamic response of shells subjected to impulsive loads The description of the spatial variation of the dynamic load is shown m l:ig. 6. The variation in 0-direction is considered to be of cosinusoidal type to limit the consideration of an excessively large number of terms in the serie~ while adequately demonstrating the solution procedure. Thus, the loading can be expressed in a single Fourier series as, q(x,O,t) =
~. q m ( t ) s i n ( m r r x / L ) c o s O
~ ].4i
m
T h e Fourier coefficients qm(t) for a centrally applied load (x~, = L/2) are given as
qm(t) = 4q°(t)sin(mrr-a/2L) m mrr
!,3,5 .
.
.
.
.
!,i
Static and transient response of cylindrical shells 0.50
~S T ' ~ ~ C ST
In= 0.05
PST,SDT.CST1
(3
0.25
171
=0.05
h
E
°
~PST, STD, CPT
-0.25 -0.50
PST, I
I
I
I
-20
40
[
~
0
I
I
20
I
-40
CST I.
I
0
-20
20
40
0.50 •
0.25
h = 0.05
PST
o
0
N
-0.25 -0.50
I
-0.02
- 0 . OZ.
-0.06
I
4
-4
0
-8
"Cxz
0.50
CST, 50T
PST"J
h = 0.15 Q
_." = o.15
025 .E N
O
CST 0 ,CST, SOT
-0.25
SDT
PST " -0.50
I
I
-8
0.50~I
I
I
I
0
-z.
~ I
I
I
I
I
-8
4
|
I
-4
0
PST h_. = 0.15
4
8 Two-Layered Shells : 0,/90 Sh¢lts Ex / E 0 = 25
Llo :,, ~:,
oI
~_5_1
E- ~-- z h / o - os shown in diQgrQms I
-O.OZ. _ -0.08
"t.'xz
1.0
I
0 5
-1.0
-2.0
Fig. 4. Comparison of through-the-thickness variation of the stresses and displacements. obtained using different shell theories, for two-layered (0/90) cylinders subjected to localized static loading.
172
A. Bhimaraddi
0.50 -
0.25 z: --.-. N
-..~OT h = 0.05 {3
CST ~ ~ . . PST
0 ST
-0.25 -0.50
• -40
I
I ::20
0
40
20
-40
-20
0.50
_,~<.~S DT
0.25 -
_
CST ~ .
~ "
- 0.50
I
I
I
-0,4
-0.2
8
DT
4
-0.25 -0.50
PST
-h- =0.15 (3
0.25
T _
i
1
-8
I
I
-4
[ 0
0 6
1
4
8
uat S D T
Ill!
~=o.,~
CST ~ ] ~ SD T--_]~L\j pST
___L____L I ~ t ~ I __.:__J -&
0
%
b_ G =O.15
0.25
.-4
./~W¢- CST /~f-~S DT
0.50 _ ~ ~ t
"-..,~
J
-0.6
"ff'x Z
- p 5 ~
~.(Z:
I
-0.25
0.50
20
0
4
Three - Layered SheHs:[ 0/90/0
Cylinder ,
Ex/ E0-25 L/ca = I
-0.25 h / G - QS ~howr! -0.50
~.
f
fl
-0.2 ~xz
I
I
-0,4
I
_L--_-i--_ 1.0
-I.0
_#o
Fig. 5. Comparison of through-the-thickness variation of stresses and displacements. obtained using different shell theories, for three-layered (0/90/0) cylinders subjected to localized static loading.
173
Static and transient response of cylindrical shells
I
~
6
×o----t oxis
,-,- !t
444--,
-I
I
Fig. 6. Description of the dynamic loading considered.
2.5
2.5~/
o
I
1 ~
l ~
I -~ =
....
u_
o
DLF Vs, To 1 oLI
lo
'
'
0/90
t/ [~ 0.5# ,
0.5
[__q ,
l
,
L/o:, I
,
,
To 20
~o
6o To
8o
Cylinder
h/o = o ,
2o
,
p, = ,1
,
E./Ee = 25 ~o
6o
8o
To
Fig. 7. Plots of dynamic load factor (DLF) as a function of the pulse duration time (To) for 0/90 cylinders subjected to rectangular pulse and for different ~/a values.
The time variation of qo(t) corresponds to the various pulse shapes shown in the diagrams. The stresses and displacements have been obtained by summing the series (14) over all m values, up to m = 301. It has been observed 3s that the dynamic load factor ( D L F - - t h e ratio of maximum dynamic response to static response) exceeded a value of 2-0 (for h- and ¥0z) for step impulsive loading and was always less than 2.0 for other impulsive loadings considered. To ascertain whether the geometry of the loading has any effect on the DLF, three different load lengths (K) have been considered and the results are shown in Fig. 7. It is evident from this figure that the D L F value depends on the geometry of the loading, this is particularly so in the case of ~, and the more concentrated the load the
A. Bhimaraddi
174
I
'3
-4ol-
...I
/
-20F
~oI
~-.
,;/
.._]
%
..j"
',,\\
¢//
1,7
...~
I 10
I
I 15
I ~'~-~I" 20
'
'
"
'
'
'
'./L :-o11 '
I
I 10
I
I 15
I 5
I
I 25
I
I __I 30
.~W
I 35
I~ 20
I
~ I~
] 40
"---'--~
\ V ' ,,//I ',~~-k-.~.:., I
/Y
'
I
-"
../I
~,",~
I 5
I
/'
,k
"'\.__..~-" I
-lop/,/
IIY
'~,, \\\
./.../
0 I~~
0 I/'
h/o: o.1
~
\~
I __3__.~_~..._a__ 25 30 35
I_ ~ . 45 ( T )
i
,..J
"'-~--~--
/-~d, uWL.' - " t LO
i __2__._.2L . . . . L5 ( T )
h/o o o Z . . . . . . ExlE8:25
# :1 %F ......... Fig. 8. P l o t o f d i s p l a c e m e n t s a n d s t r e s s e s as a f u n c t i o n o f t i m e { T), o b t a i n e d u s i n g d i l f c r c m s h e l l t h e o r i e s , for a 0/90 c y l i n d e r . - P S T (Tp = 19"8144), -- S D T (I~, .... t 9 ' 8 2 4 '7 i. ..... C S T (Tp = t9.7(}88).
h/a
-- 0,15
-4O -
30
'3 -
20 -10 5
10
15
20
25
h/o = o. 1~
30
35
40
45
/"
(T)
b("
-2O -15 10 -5 5
I0
15
20
25
30
35
40
h/a=O.IS E x / E 0 = z5
&5
(T}
LI~=I ~ . I
/ca : 1/4 q o ~ _ _
F i g . 9. P l o t o f d i s p l a c e m e n t s a n d s t r e s s e s as a f u n c t i o n o f t i m e ('/3, o b t a i n e d u s i n g d i f f e r e n t s h e l l t h e o r i e s , f o r a 0/90 cylinder. - P S T , - . . . . . . S D T , -. . . . . . . CST,
Static and transient response of cylindrical shells
175
I ,3
o.L / ~ -
I
_
"\
",.-X----~
".. I
5
I
I
10
I
l
I
15
~.~---'-~
\
"7"-,.
I
I
20
/
~'~,,/
~
7--" / ' m 7~,
I
I
25
30
I
I
35
I
~
,
",~-'~.~ ,, ---f I
I
.40
I
45
I
(T)
-30 -20 i b x _ 10 -
0 10
,..
\
.
F i g . 1 0 . P l o t o f d i s p l a c e m e n t (~) and stress (~-x) as a function o f time (T), o b t a i n e d for d i f f e r e n t pulse s h a p e s , for a 0/90 cylinder (h/a = 0.1, L / a = 1,/3 = 1. ~la = ~, Exl Eo = 25, T o - 19"814).
higher is the DLF value. However, the DLF value did not exceed 2-0 in the case of ff~, crx and fro. Figures 8 and 9 show the variation of ~ and ~Txwith time (T) obtained using various shell theories. We notice from these figures that the variation of the response curves is periodic with a period almost equal to the longest natural period (values are shown), corresponding to the first flexurai mode of vibration. The irregular or zig-zag nature of these curves indicates the contribution of the higher harmonics associated with the localized nature of the loading. Also, the peak values of ~ predicted by SDT are high when compared with PST, whereas those of CST are lower. The maximum f~ response predicted by CST and SDT is always less than that predicted by PST. Figure 10 shows the response curves obtained using PST for various pulse shapes. The pulse duration time has been taken to be equal to the first natural period (Tp = 19"8144). It is clear that the maximum response occurs during the application of the loading and the response curves for the symmetric triangular impulse and the sine-wave impulse follow similar trends. They are almost identical in the free vibration region ( T > To). The response curves of decreasing and increasing triangular impulses are out of phase with each other in the free vibration region.
176
,4. Bhirnaraddi 5 CONCLUSIONS
In conclusion, we note that the stress and displacement distribution across the thickness of the shell is nonlinear and this nonlinearity increases with the increase in thickness and in the material orthotropy, and with the decrease in the localized load dimensions. The linear stress and displacement distributions of the classical and the shear deformation theories diffel: considerably from the nonlinear distribution predicted by the proposed shell theory. The dynamic load factor can assume values greater thal~ 2.{) depending upon the load geometry.
ACKNOWLEDGEMENTS Financial assistance, in the form of a writing award, was provided by the Faculty of Engineering, University of Melbourne, Australia. In addition~ the author thanks Professor L. K. Stevens and the referee for their c o m m e n t s and suggestions.
REFERENCES l. Donnell, L. H., Stability of Thin Walled Tubes in 7k)rsion, NACA Report No. 479, 1933. 2. FlLigge, W., Stresses in Shells, Springer-Vcrlag, Berlin, 1962. 3. Love, A, E. H., Treatise on the Mathematical Theory of Elasticity, Dover, Nc~ York, 1944. 4. Gol'denveizer, A. L., Theory c~fElastic Thin Shells, Pergamon Press, Oxford, 1961. 5. Sanders, J. L. Jr, An Improved PTrst-Approximation Theory.fi~r Thin Shells. NASA TR R-24, June 1959. 6. Reissner, E. and Tsai, W. T., Pure bending, stretching, and twisting of laminated anisotropic shells, Trans, ASME, J. Appl. Mech., 39 (1972) 148-54~ 7. Bert, C. W., Analysis of Shells, in Structural Design and Analysis, Part t (Chamis, C. C., ed.), Vol. 7 of Composite Materials (Broutman, L. J. and Krock, R. H., eds), Academic Press, New York, 1974, Chapter 5,207-58. 8. Greenberg, J. B. and Stavsky, Y., Vibrations of laminated filament wound cylindrical shells, A I A A J., 19 (1981) 1055-62. 9. Bijlaard, P. P., Stresses from radial loads in cylindrical pressure vessels, The Welding J., 33 (1954) 615s-23s. 10. Vlasov, V. Z., Basic Differential Equations in General Theory o]'Elastic Shells.. NASA TM 1241, 1951. 11. Mirsky, I. and Herrmann, G., Nonaxially symmetric motions of cylindrical shells, J. Acoust. Soc. Am., 29 (1957) 1116-24.
Static and transient response of cylindrical shells
177
12. Naghdi, P. M., On a variational theorem in elasticity and its application to shell theory, Trans. ASME, J. Appl. Mech., 31 (1964)647-53. 13. Zukas, J. A. and Vinson, J. R., Laminated transversely isotropic cylindrical shells, Trans. ASME, J. Appl. Mech., 38 (1971) 400-7. 14. Dong, S. B. and Tso, F. K. W., On a laminated orthotropic shell theory including shear deformation, Trans. ASME, J. Appl. Mech., 39 (1972) 1091-7. 15. Greenberg, J. B. and Stavsky, Y., Vibration of axially compressed laminated orthotropic cylindrical shells, including transverse shear deformation, Acta Mech., 37 (1980) 13-28. 16. Whitney, J. M. and Sun, C. T., A refined theory for laminated anisotropic, cylindrical shells, Trans. ASME, J. Appl. Mech., 41 (1974) 471-6. 17. Gamby, D. and Bourdillon, H., Nonaxially symmetric thermal stresses in thick cylindrical shells, Trans. ASME, J. Appl. Mech., 50 (1983) 217-18. 18. Hildebrand, F. B., Reissner, E. and Thomas, G. B., Notes on the Foundations of the Theory of Small Displacements of Orthotropic Shells, NACA TN 1833, 1949. 19. Naghdi, P. M., On the theory of thin elastic shells, Quartly Appl. Mathe., 14 (1957) 369-80. 21). Miller, R. E. and Boresi, A. P., Strain energy expression for a circular cylindrical shell including transverse shear effects, Proc. 7th Midwestern Mech. Conf., Michigan State Univ., Sept. 6-8 1961, Developments in Mechanics, Vol. 1, J. E. Lay and L. E. Malvern, eds, North-Holland Publishing Co., Amsterdam, 1961,341)-54. 21. Mirsky, I., Vibration of orthotropic, thick, cylindrical shells, J. Acoust. Soc. Am., 36 (1964) 41-51. 22. Nikolai, R. J., Boresi, A. P. and Ford, J. L., A theory for the anisotropic axisymmetric elastic shell with application to the orthotropic cylinder, Proc. 4th Southeastern Conf. Theoretical and Applied Mechanics, Tulane Univ., New Orleans, Louisiana, Feb. 29-Mar. 1 1968. Developments in Theoretical and Applied Mechanics, Vol. 4, Daniel Frederick, ed., Pergamon Press, 19711, 199-220. 23. Golub, E. B. and Romano, F., A method for obtaining stresses and displacements in thick cylindrical shells under arbitrary boundary conditions, Trans. ASME, J. Appl. Mech., 40 (1973) 221-6. 24. Benzley, S. E., Hutchinson, J. R. and Key, S. W., A dynamic shell theory coupling thickness stress wave effects with gross structural response, Trans. ASME, J. Appl. Mech., 40 (1973) 731-5. 25. Lo, K. H., Christensen, R. M. and Wu, E. M., A high-order theory of plate deformation, Pt-l: homogeneous plates, Trans. ASME, J. Appl. Mech., 44 (1977) 663-8. 26. Widera, G. E. O. and Logan, D. L., Refined theories for nonhomogeneous anisotropic cylindrical shells; Part I: derivation, Proc. ASCE, J. Engng Mech. Div., 106 (1980) 1053-74. 27. Logan, D. L. and Widera, G. E. O., Refined theories for nonhomogeneous anisotropic cylindrical shells; Part II: applications, Proc. ASCE, J. Engng Mech. Div., 106 (1980) 1075-90. 28. Levinson, M., An accurate, simple theory of the statics and dynamics of elastic plates, Mech. Res. Commun., 7 (1980) 343-50.
178
A. Bhirnaraddi
29. Bhimaraddi, A. and Stevens, L. K., A higher order theory for free vibration ot orthotropic, homogeneous, and laminated rectangular plates, Tram. ASME~ J. Appl. Mech., 51 (1984) 195-8. 30. Reddy, J. N., A refined nonlinear theory of plates with transverse shear deformation, Int. J. Solids and Structures, 20 (1984) 881-96. 31. Reddy, J. N., A simple higher-order theory for laminated composite piate~, Trans. ASME, J. Appl. Mech., 51 (1984) 745-52. 32. Bhimaraddi, A. and Stevens, L. K., On the higher order theories in piatc~ and shells, Int. J. Struct., 6 (1986) 35-50. 33. Bhimaraddi, A., A higher order theory for free vibration analysis of circtflar cylindrical shells, Int. J. Solids" and Structures, 20 (1984) 623-3(i. 34. Bhimaraddi, A., A higher order theory for dynamic response of orthotropic homogeneous and laminated cylindrical shells, AIAA J., 23 (1985) 1834 7 35. Armenakas, A. E., Gazis, D. and Herrmann, G., Free Vibration of ('itcular Cylindrical Shells, Pergamon Press, Oxford, t969. 36. Bathe, K.-J. and Wilson, E. L., Numerical Methods" in Finite Element Amth:sc~. Prentice-Hall of India Pty Ltd, New Delhi, 1978. 37. Newmark, N. M., A method of computation for structural dynamics, ,,"to, ASCE, J. Engng Mech. Div., 85 (1959) 67-94. 38. Bhimaraddi, A., Static and dynamic response of plates and shells. PhD 'l'hcsls~ Department of Civil Engineering, University of Melbourne, February 1985 39. Timoshenko, S. and Goodier, J. N., Theorv of Elasticity, 2nd edn, McGra~Hill, New York, 1951.
APPENDIX Definitions of 1~ appearing in eqns (8)
,1,4,7] 12 15 /~ 13 16 19
;
[,,o,,, 113,1 1 ,,4 ,,7/ =f 112 I15
(ai
z2
(a+z)~,
, dz aq- Z a ¢-Z
ib
I18.]
119120] 134 135
,]
(I,a + z,$)dz
=
z I (l,~)dz
(a+ z)2I a+t
to)
Static and transient response of cylindrical shells
121= y
(d)
(2(a+z)3dz
(122"123'124) = y
179
( ~ Z ) (Z2'z4'~2)dZ
(e)
126 I~
127 159
=f
z(2a+z)
a
+Z,
1
a+z
dz
(f)
144 142
[ 14° ]
1
= y Z2(2a+z)
/ 30
dz
2a+ z
LV~:.
2,331 , °+~,]
L l2_s 141
(g)
136
~2
(l,a + Z,(a + Z)")dz
(h)
(a + z) 2
(z, z~)dz
[39 [38
(2a + c)
(i)
Static and Transient Response of Cylindrical Shells
A. Bhimaraddi* Department of Civil Engineering, University of Melbourne. Parkville. Victoria 3052~ Australia (Received 20 April 1985; revised version accepted 27 August 1986)
A BS TRA CT Static and transient analysis ofcomposite ~Tlindrical shells" is presented using a recently proposed shear deformation theory. The dynamic response is obtained by employing the numerical time integration scheme due to Newmark. The results obtained by using classical shell theory ( ( S T ) and Mindlin-type shear defbrmation theory (SDT) are compared with those obtained by using the proposed theory. The comparison studies reveal that the linear stress distribution, as assumed in CSTand SDT, differs considerably from the predicted nonlinear distributhm qC'theproposed theory.
NOTATION a,L -~-~ Ex, Eo, Ez G xo, G x~, G ~z h
hl, h2, h3 ITl, Fl
Mean radius and length of cylinder. Localized load dimensions (see Fig. 1). Young's moduli. Shear moduli. Total thickness of shell. Thicknesses of the individual layers. Axial (half) and circumferential wave numbers.
* Present address: Department of Civil Engineering, University of Canterbury. Christchurch l, New Zealand. 157
Thin-Walled Structures 0263-8231/87/$03.50 © Elsevier Applied Science Publishers Ltd, England, 1987. Printed in Great Britain
158
.4. Bhimaraddi
N~,No, N~o, No~ } M~, Mo, Mxo, Mo~ Stress-resultants. M '~, M 'o, M 'xo,M 'o~ Applied loads on the outer and the inner surface ot the qzo, qz~ cylinder in the z-direction~ t Time coordinate. Pulse duration time. to tp Fundamental natural period (= 2n-/~)o LI, V, W Displacement components in the x-, O- and z-directi~ m s Generalized displacement quantities. U0, /~1. ]20, Vl, W0 x,O,z Cylindrical coordinate system. Relative thickness ratios in laminated cylinders [3, = h~_/h,,[33 = h2/h3,[3 = [3~ :----/3:,. Shear strains in cylindrical coordinates. ]/xo, 3/xz, "Yoz 6~ Percentage error in classical shell theory. Percentage error in shear deformation theor}.. At Time step-size used in numerical time integration. Normal strains in the x-. 0- and z-directions. Ex, ~-o~ Ez Poisson's ratio. I~ ~o Angular frequency. P Mass density of the material. Mass densities of individual layers. pl, p2, p3 Normal stresses in the x-, 0- and z-directions. O'x, C1"O~O" z Shear stresses in cylindrical coordinates. "grO~ T~z, TOz
( To, T, AT, Tp) =:
(10, t, ~,1, t,,)
L x
(a,F,~) = - - ( u , v , w ) aqo l
q0
1 INTRODUCTION The increased use of composites for high performance design applicanons necessitates more realistic prediction of the response characteristics of composite structures. It is possible to treat composite shells constructed of laminates by applying three-dimensional elasto-dynamics to each individual
Static and transient response of cylindrical shells
159
layer in conjunction with appropriate continuity and boundary conditions. However, it is well known that the complexities involved in the mathematical treatment of laminated shells using three-dimensional elasticity make it impractical to carry out such an analysis. To avoid this complexity, it is the usual engineering practice to make certain assumptions regarding the kinematics of deformation in the thickness direction of shell. For example, in the classical shell theories (CST)I-9 it is assumed that normals to the midplane before deformation remain straight and normal to the midplane after deformation; the implication is that the thickness shear deformations are negligible. Recent developments in the analysis of plates and shells laminated of fiber-reinforced materials indicate that thickness has more pronounced effects on the behaviour of composite plates and shells than on isotropic laminates. Also, due to low transverse shear moduli relative to the in-plane Young's moduli, thickness shear deformations are even more pronounced in composite laminates. Hence, a mathematical model to analyse composite laminated shells, in order to be closer to reality, must incorporate the effects of transverse shear deformation. There exist many shell theories that account for transverse shear strain effects.~°-~5 These shear deformation theories (SDT) are based on the assumption that the normals to the midplane before deformation remain straight but no longer remain normal to the midplane after deformation; the implication is that the in-plane displacement components are assumed to be linear functions of the normal coordinate (z) and warping of cross-sections is not allowed. For accurate determination of response characteristics of shells, in cases such as when accurate higher order frequencies are required 16or when there exists an appreciable thermal gradient through the thickness, ~7the above mentioned shear deformation theories are found to be inadequate. Thus, there have appeared many higher order theories ~6-27which account tor transverse deformations and warping of cross-sections. Usually, higher order theories are based on the assumed form for displacements--taken as power series expressions in z. It may be stated here that most of the available higher order theories employ too many unknown displacement parameters in order to achieve parabolic variation for transverse shear strains. In addition these theories, with the exception of ref. 17, do not satisfy shearfree conditions on the outer and the inner surfaces of the shell. Further discussions on the applicability and the accuracy of such formulations for plates and shells can be found in ref. 25. The higher order plate theories in which in-plane displacement components are assumed as cubic polynomials in z, resulting in parabolic variation for transverse shear strain distribution, can be found in refs 28-31. These theories employ fewer unknown displacement parameters, equal to
160
A. Bhimaraddi
those used in shear deformation theories, 11.15and they also satisfy shear-free surface conditions. However, it is to be noted here that, owing to the difference in the assumed in-plane displacement forms, classical plate theory equations can be obtained as a special case from ref. 29 but not from refs 30 and 31. Also, the displacement forms chosen in ref. 29 are more versatile than those of refs 30 and 3 l, in a sense that almost any {:(z) (see eqn (21 below) whose first derivative vanishes at +-h/2, and is non-zero elsewhere. can be used in ref. 29, resulting in a corresponding higher order theory. Such various forms for sO(z) and the procedure for systematic reduction ot equilibrium equations in terms of stress-resultants for plates and shells using three-dimensional stress equilibrium equations have been given in ref. 32, A higher order theory for cylindrical shells, similar to that for plates. > has been given by the author 33'34in the context of free vibration analysis. The superiority of the proposed shell theory (PST) has been demonstrated in re[. 33, by comparing the results obtained by PST, SDT and CST with those obtained by using three-dimensional theory of elasticity, 3s in the context ot free vibration analysis of isotropic cylinders. A similar comparison lk~r orthotropic homogeneous and laminated shells has been given in ref. 34. In this paper the complete formulation of the theory, for generally orthotropic shells and also further comparison of PST, SDT and CST in the context ol static and forced vibration analysis has been dealt with. A solution to the forced vibration response has been obtained by numerical integration of the resulting system of dynamic equilibrium equations using Newmark~ method. 36,37
2 G O V E R N I N G E Q U A T I O N S OF MOTION The components of displacements are assumed as Mllows (refer to the Notation) u (x, 0, z, t) = u0(x, 0, t) + {:ul(x, 0, t) - ,'.w0, ~ a-Z~ Wl,, 0 w ( x , O, Z, t) = Wo(X, O, t)
where : z
( 4z2/ l-3-
~tl
Static and transient response o f cylindrical shells
16 1
In these equations a comma denotes differentiation with respect to the letter(s) followed by it. By writing out the expressions for strains 32'3339using the above displacement components one can easily verify that the transverse shear strains, and hence stresses, vanish on the outer and inner surfaces of the cylinder and that they vary parabolically across the thickness of the shell. This obviates the use of shear correction factors, usually employed in Mirsky-Herrmann type" shear deformation theories. It may be said here that the form of in-plane displacement components in eqns (1) is not the only choice. There could be other possible alternatives to choose for ((z) to accomplish the above mentioned purpose. The other forms for ((z) may be found in ref. 32. Also, if one wants to have a greater number of terms in the w-expression, corresponding terms should be added to the u- and v-expressions to have the shear-free surface boundary conditions satisfied. This will, of course, result in a more accurate but more complicated theory involving the consideration of thickness normal strain effects, and a greater number of generalized displacements. Hence, to limit the complexity of the problem to a reasonable degree, the w-expression has been chosen to be a constant across the thickness of shell. The constitutive relations corresponding to a monoclinic type of material (or for any layer in the case of a laminate) are of the form, (r~ = (_'lte.~+ Cl2Eo+ C~ey~o ; cro = CI2E~+ C22~e+ C26y.o
(3)
"F~0 = CI6Ex+
C26Eo + C66")lxo ; rxz = C44Txz + C45Toz ; for = C45")/xz+ C55Toz
We have the following definitions for stress-resultants appropriate to the present method of analysis 32'343~ =
(or,, o-0) .___5
M i~ M'o
Nxo M.'~o
= f
It'] o+Z/2 --7]
(l'sC)rx°dz
(4)
a Q . = aMx, x + (M.o + M~x), o + PsUo, t, + p llu 1, . - - p6Wo ....
(5) a2Qo = aMo. o+ aZ(Mxo+ M o . ) , ~ + (apS + p 6 ) v o , . + (ap ~) + p~2) v l , . -
p6wo. o.
162
,4. Bhimaraddi
and the equilibrium equations in terms of stress-resultants can be writter~ as,
32
aN ..... + No~. o = - a y
pu, u ( - - /+s z
dz
aN~o, .~+ No. o = - a I
pv, t, (--~-[--)a+ z
-dz
aM', ~+M'ox, o - a
aM'o,~+M'o,,--a
~'xz~l a * c ,,
.r
j
roz~l --~-Z
1
aM ..... + ( M ~ o + M ~ ) , x o + - M ~,oo+ a p(azu, ,.,,+ zv. o, +
,,) (
I-<.-)
dz ...... a
[u,,,~'i~"\d: t
dz = -a
f
y
Ov,,,~\
+~-- dz
Cr0dz .... q
dz
where ~ = ~,= =
4.,,: 1-~-)
h ; q = a(q:,, - %,I t - ~ ( q : o . q : , )
H e r e , q zoand q z~are the applied loads in the z-direction on the outer and in nc r surfaces of the shell, respectively. Substitution of stress-resultants (41. expressed in terms of the generalized displacements of the problem, in eqn~ (6) results in the following set of dynamic equilibrium equations~ L8
=
P
where 8 + = (Uo, Vo, U . V . W o ) pT = (O,O,O,O,q)
and the operators L u are symmetric and have the following forms:
163
‘L,
.I L13
L.32
=
L2.1
a2c:
il = .J1 J u2cg
-$().
+
c7422
Ml +
c:;
‘I
1
I?
L,4
f( 1, ,x+
L 23
Li.4
.
+
c:,
Cih
0
0
1
J( 1.HH-
c-i_,
0
c-4126
_
c$ + cg.j [ cy,+ cz
I
C::
A(19
XH”
1(}4
A. Bhimaraddi
l
[]
a { ).o~o+
'~:
( }.,-'
(";2 ]
( >
/
a-
L~s : -~-'?(aeC'~:() ..... +
-,~
Ii1
_
J
( )-o-
I r,l~,
[1
L
'
L " 2~,
P ['
2aC~{ ), ,,,o + (2a("}2 r {-3,~
~-2
"aC '~
a~'('~]{
}
+a2p~( ) .... ,+P6( ).oo,,_ a 2 p4 { )..)
L45 ]
C iB
I
c.+
~'~ 26
llii1
a-
c~2
1
4-
g 44
I a-Cil v ,251
{~,;,
'
~{ ).,, . . . . . .
L c L-J
[ L~3]
~g
'
.-:{ }.,,,~
, -'? [a(
t
'46,1"1
a - ( ;g 1 -:( i
2
L c~ J
tc~J
a"(44
1
I ~{
a2{ ) +
L c~
),,,
o :1
w h e r e . ( ) ..... etc., r e f e r to derivatives o f the c o l u m n matrix o f the g e n e r a l i z e d d i s p l a c e m e n t s s h o w n in e q n (7) a n d , cikl = IkCij ; pk __ Ikp
:~)
Staticand transientresponseof cylindricalshells
165
where Ik are the various integrals to be evaluated between the limits - h / 2 to + h / 2 , as defined in the Appendix. The boundary conditions along the edge of the shell require that either one member of each of the following six pairs, or six linearly independent combinations of them, must be specified, Along x = constant: N,u() ; Nxovo ; Mi~ui ; Mi~oVl ; Qxwo ; Mxwo, x Along 0 = constant: Nod~() ; N~vo ; M'oxul ; M',v, ; Qowo ; Mowo, o
The governing equations of the classical shell theory can be obtained by deleting the third and fourth rows and columns, which correspond to u~ and v~ displacdlnent parameters, from the matrices of eqn (7). Similarly, all o t h e r relations, such as stress-resultants, boundary, conditions, etc., can be obtained by ignoring the terms containing u~ and v ~.
3 SOLUT1ON OF T H E E Q U I L I B R I U M E Q U A T I O N S The closed-form solutions to the equilibrium eqns, (7), with generally orthotropic (monoclinic) material are difficult to obtain. However, specially orthotropic, simply supported shells render possible the closed-form solution in the form of a double Fourier series, each term of which is explicitly defined by a set of simultaneous equations. The shell is assumed to be simply supported at x = 0 and x = L and the solution is assumed to be ,,,)(x.o,t)=
T. n
.,(x.o,t)
=
Vo(X,o,t) rn
-T.
v,,,..(t),b..,,,(x.o) n
E Olmn(t)(l)lrnn(x,O); Vl(X,O,l)~ ~ n
w0(x. o, t) :
m
n
m
~ Vlmn(t)(Ik2mn(x,O) rn
(9)
E W°,nn(t)OP3mn(X, O) n
m
where (~) Iron
= cos(mTr.r/L)cos(nO)
" ~2,,~ = sin(rnrrx/L)sin(nO)
qb3m, = sin ( m r r x / L ) cos (nO) The spatial variation of the loading has been assumed in the form of a double Fourier series to be q(x,O,t) =
~
~
n
m
qm,(t)sin(mrrx/L)cos(nO)
(10)
A. Bhimaraddi
166
where the terms q~), in the above series can be determined as qm=(t) = 0
q,~(t)
= ~
, m = n =0
1 y'"
qm,,(t) = ~ -
q(x.O,t)dO
IJ" .
m :~: (I. n ..... i)
q(x,O,t)sm(mrrx/L)cos(nO)d.vdO
~
~
., ,~ 4= !~ t~. j ,~
Substitution of eqns (9) and (10) in eqns (7) results in the following system of second order ordinary differential equations, variable in time, for each se~ of ( m , n ) values, MA+KA- q
~!!)
where K and M are the symmetric stiffness and mass matrices of order 5 and we have the following definitions for A. A and q,
A =
"
Uornn, V(Imn, UI .... Vlmn" W,)rnn l h
q
=
tO,O,O,O,(a+~)q,,~(t)}
Note that the superposed dots indicate differentiation with respect to troy.:. For a given set of (rn, n) values the A vector can be obtained at vari~ms desired values of time by solving eqns (12) using the time integration scheme due to Newmark. 36.37By summing the series over all (m, n) values up to the desired degree of accuracy, displacement and stress distribution in space and time can be obtained for a specified variation of the external loading m space and time. However, for a static problem the A vector is computed by solving the set of algebraic equations which are obtained from eqns (12) by ignoring the time dependence of displacements and loads. Lastly, it should be said here that the transverse shear stress distribution across the thickness, in the case of SDT and CPT, has been obtained by lhst computing the values of shear forces and then distributing them according to the parabolic law. However, the actual shear stress distribution is constant according to SDT and is zero according to CPT. The spatial location oi the
Static and transient response of cylindrical shells
167
points to which the values of various displacements and stresses (referred to in the discussion of numerical results) correspond, is as given below,
( w , ~ . g / o )
at
at
x L
-
z h
-
0 7r
x L
at
x
I 2'
-
-
0
0
,
=
0
z h
-
z h
-
0 rr
-
1 2
z h
-
0 7r
-
0
L
~z
at
x L
-
to..
at
x L
-
0 rr
1 2'
-
x ~o at ~ - = 0
0 n-
1 2
-
0
: h z
'
-
1 -
h
2
4 DISCUSSION OF NUMERICAL RESULTS In all of the numerical results presented herein, zero initial conditions fl)r displacements and velocities have been assumed. The numerical results for transient response have been obtained by using a time step-size (AT) equal to 1% of the predominant natural period of the system. The accuracy and the correctness of the algebra and the computer coding have been ensured by comparing the results with those available in the literature. 9 The S D T results in the present study have been obtained by using a shear factor value of ~.2/12. There are different shear factor values available in the literature for homogeneous and laminated plates and shells which have often been obtained for a particular problem under consideration. Hence, it b e c o m e s extremely difficult to choose these values for the problems dealt with in this paper. Usually, a value of 7r2/12 is used in the absence of a rational explanation as to the use of other values than 7r2/12. Further discussion on the use of shear factor values may be found in ref. 38. Detailed studies on the static and dynamic response of cylinders with various geometric and material parameters were undertaken. 39 However,
168
A. Bhimaraddi
only a few results have been given here for the sake of brevity. The influence of material orthotropy has been considered using E x / E o values of 1.25 and 40 while keeping G~o = Gx~ - G ~ : '._,Eo, tz~, = '. p~ .... P2 : P~ :- P. 4.1 Static r e s p o n s e o f a shell subjected to localized loading
For a localized static load of intensity q0 applied over an area a x {~ with tile centre at x = x0 and 0 = 0, as shown in Fig. 1, q~. is given by q.,. = 0
. m = n = 0
a + - ~h/ '~ qm.
qmn
_
=
2bq0 mTr2Sin(mTr-a9 / - L ) :si n ( rn7rx o/L). tn 4= q}. n - - ( )
8q0 s i n ( m T r ~ / 2 L ) s i n ( m r c x o / L ) s i n
7.g2mn
[ ,,
•
m ~ (). t~ i
~i
In the numerical results presented here a centrally applied localized load of square shape (~ = b; xo = L / 2 ) has been considered. Figures 2 and 3 show the influence of h / a ratio on radial displacement ( ~ ) and axial stress ( ~ j . The percentage differences (6c in CST and 6. in SDT) have t~cen c o m p u t e d as follows, 6c=
value from CST ) value fromPST - 1 ×100
6~:
value from SDT ) value fromPST - 1 × l(~l
-~ b ~
"~-x°--W axis
X
L
Fig. 1. Description of the localized static loading.
Static and transient response of cylindrical shells
- - 6 s
Ex /EB= 25
.... Sc Ex/Ee = 25
L l e = 1.0 8
O=l.O b
-0.6
I
6
4 -0.4
m o g e L . . . . _ _
~
¢,
-20
\oj,.-
/
// /I
2
0/90 ..
~%~/ (b'/~(3 .
_ b _ 1 ~./~p) ' / ~--~--~ .%, / e°,; " ,~I II 0\/
-30
x:lo
/
\ ~I I
Lie -- !.0
-40
-0.5 ,3
169
/I
/~+,/~,J
~.
-10
-0.3
0 ,
0.05
0.10
O.05
0.15
O.10
O.15
h/a
h/~
Fig. 2. Variation of radial (~) displacement and the corresponding percentage differences in SDT ((5~)and CST (8~) for different thickness-to-radius (h/a) ratios of cylindrical shells under localized static loading (Homoge. = homogeneous).
---
~(I)~10
E,/E0o 25 L/~
-0,2C
: 1,0
/~/
/3~c~y//.
p:l.O
P -40
-010
-012
x"/ / / /
///
,S s
------
Sc
Ex / E 8 : 25 L/a -- 1.0
13:1.0
~_i_i ~-~-~
~,+~
-30
.o~"~ F / " .i j~×
- -
..-//+o¢ ~o\~
ILO
/o\"
-20
-0.08
//'./
• /
//// -0.0~
V, o!
-10
,'/..S..;
/
>X~U'_~
~-
/ I
I
}
0,05
0.10
0.15
h/a
0.05
0.10
0.15
h/o
Fig. 3. Variation of normal (~x) stress and the corresponding percentage differences in SDT ( S j and CST (8c) for different thickness-to-radius (h/a) ratios of cylindrical shells under localized static loading.
,4. Bhirnaraddi
170
Also, due to its superior performance (to CST and SDT) in the case ot h o m o g e n e o u s isotropic shells when compared with elasticity results, 3~'~ ~s and due to the nonavailability of elasticity results for composite shells, comparison of various shell theories has been made taking PST as vhe standard solution. It may be seen from Fig. 2 that the ~ response for homogeneous shetis is high c o m p a r e d to that of layered shells. Also, if: values for three-layL:red shells are less than those for two-layered shells, except for h/a < 0"05. t'ot which the 0/90/0 shell has higher ~ values than two-layered shells. Further. we observe that CST underestimates (8~.<:0) ~ values, whereas S D I overestimates (8~>0) these values. The 8~ and 8~ values increase with increasing h/a. The maximum value of 8~ is about - 5 0 % and that ol #5, is about + 7 % as may be seen from Fig. 2. These values will be still higher h~l higher values of E~/Eo and for more concentrated loadings. From Fig. 3(left) we note that the stress response is maximum Io~ h o m o g e n e o u s shells as compared to layered shells. From Fig. 3(right) we note that both CST and SDT underestimate axial stress value when c o m p a r e d with PST. The errors increase with increasing h/a. with a m a x i m u m value of - 3 5 % for h/a = I). 15. These errors will be still higher for higher h/a and E,/Eo values and the more concentrated the load becom,=s. It is evident from Figs 4 and 5 that as the thickness of shell increases the normal stress and the in-plane displacement variations across the thicki~csof shell b e c o m e more and more nonlinear. It has also been observed thal t~)[ higher E,/Eo ratios, for more concentrated localized loads, and for small L/a values, stress and displacement distribution becomes more and m,~,rc' nonlinear. Hence, for such parameters the predictions based on the lfi,cm distribution, as assumed in CST and SDT, differ considerably.
4.2 Dynamic response of shells subjected to impulsive loads The description of the spatial variation of the dynamic load is shown m l:ig. 6. The variation in 0-direction is considered to be of cosinusoidal type to limit the consideration of an excessively large number of terms in the serie~ while adequately demonstrating the solution procedure. Thus, the loading can be expressed in a single Fourier series as, q(x,O,t) =
~. q m ( t ) s i n ( m r r x / L ) c o s O
~ ].4i
m
T h e Fourier coefficients qm(t) for a centrally applied load (x~, = L/2) are given as
qm(t) = 4q°(t)sin(mrr-a/2L) m mrr
!,3,5 .
.
.
.
.
!,i
Static and transient response of cylindrical shells 0.50
~S T ' ~ ~ C ST
In= 0.05
PST,SDT.CST1
(3
0.25
171
=0.05
h
E
°
~PST, STD, CPT
-0.25 -0.50
PST, I
I
I
I
-20
40
[
~
0
I
I
20
I
-40
CST I.
I
0
-20
20
40
0.50 •
0.25
h = 0.05
PST
o
0
N
-0.25 -0.50
I
-0.02
- 0 . OZ.
-0.06
I
4
-4
0
-8
"Cxz
0.50
CST, 50T
PST"J
h = 0.15 Q
_." = o.15
025 .E N
O
CST 0 ,CST, SOT
-0.25
SDT
PST " -0.50
I
I
-8
0.50~I
I
I
I
0
-z.
~ I
I
I
I
I
-8
4
|
I
-4
0
PST h_. = 0.15
4
8 Two-Layered Shells : 0,/90 Sh¢lts Ex / E 0 = 25
Llo :,, ~:,
oI
~_5_1
E- ~-- z h / o - os shown in diQgrQms I
-O.OZ. _ -0.08
"t.'xz
1.0
I
0 5
-1.0
-2.0
Fig. 4. Comparison of through-the-thickness variation of the stresses and displacements. obtained using different shell theories, for two-layered (0/90) cylinders subjected to localized static loading.
172
A. Bhimaraddi
0.50 -
0.25 z: --.-. N
-..~OT h = 0.05 {3
CST ~ ~ . . PST
0 ST
-0.25 -0.50
• -40
I
I ::20
0
40
20
-40
-20
0.50
_,~<.~S DT
0.25 -
_
CST ~ .
~ "
- 0.50
I
I
I
-0,4
-0.2
8
DT
4
-0.25 -0.50
PST
-h- =0.15 (3
0.25
T _
i
1
-8
I
I
-4
[ 0
0 6
1
4
8
uat S D T
Ill!
~=o.,~
CST ~ ] ~ SD T--_]~L\j pST
___L____L I ~ t ~ I __.:__J -&
0
%
b_ G =O.15
0.25
.-4
./~W¢- CST /~f-~S DT
0.50 _ ~ ~ t
"-..,~
J
-0.6
"ff'x Z
- p 5 ~
~.(Z:
I
-0.25
0.50
20
0
4
Three - Layered SheHs:[ 0/90/0
Cylinder ,
Ex/ E0-25 L/ca = I
-0.25 h / G - QS ~howr! -0.50
~.
f
fl
-0.2 ~xz
I
I
-0,4
I
_L--_-i--_ 1.0
-I.0
_#o
Fig. 5. Comparison of through-the-thickness variation of stresses and displacements. obtained using different shell theories, for three-layered (0/90/0) cylinders subjected to localized static loading.
173
Static and transient response of cylindrical shells
I
~
6
×o----t oxis
,-,- !t
444--,
-I
I
Fig. 6. Description of the dynamic loading considered.
2.5
2.5~/
o
I
1 ~
l ~
I -~ =
....
u_
o
DLF Vs, To 1 oLI
lo
'
'
0/90
t/ [~ 0.5# ,
0.5
[__q ,
l
,
L/o:, I
,
,
To 20
~o
6o To
8o
Cylinder
h/o = o ,
2o
,
p, = ,1
,
E./Ee = 25 ~o
6o
8o
To
Fig. 7. Plots of dynamic load factor (DLF) as a function of the pulse duration time (To) for 0/90 cylinders subjected to rectangular pulse and for different ~/a values.
The time variation of qo(t) corresponds to the various pulse shapes shown in the diagrams. The stresses and displacements have been obtained by summing the series (14) over all m values, up to m = 301. It has been observed 3s that the dynamic load factor ( D L F - - t h e ratio of maximum dynamic response to static response) exceeded a value of 2-0 (for h- and ¥0z) for step impulsive loading and was always less than 2.0 for other impulsive loadings considered. To ascertain whether the geometry of the loading has any effect on the DLF, three different load lengths (K) have been considered and the results are shown in Fig. 7. It is evident from this figure that the D L F value depends on the geometry of the loading, this is particularly so in the case of ~, and the more concentrated the load the
A. Bhimaraddi
174
I
'3
-4ol-
...I
/
-20F
~oI
~-.
,;/
.._]
%
..j"
',,\\
¢//
1,7
...~
I 10
I
I 15
I ~'~-~I" 20
'
'
"
'
'
'
'./L :-o11 '
I
I 10
I
I 15
I 5
I
I 25
I
I __I 30
.~W
I 35
I~ 20
I
~ I~
] 40
"---'--~
\ V ' ,,//I ',~~-k-.~.:., I
/Y
'
I
-"
../I
~,",~
I 5
I
/'
,k
"'\.__..~-" I
-lop/,/
IIY
'~,, \\\
./.../
0 I~~
0 I/'
h/o: o.1
~
\~
I __3__.~_~..._a__ 25 30 35
I_ ~ . 45 ( T )
i
,..J
"'-~--~--
/-~d, uWL.' - " t LO
i __2__._.2L . . . . L5 ( T )
h/o o o Z . . . . . . ExlE8:25
# :1 %F ......... Fig. 8. P l o t o f d i s p l a c e m e n t s a n d s t r e s s e s as a f u n c t i o n o f t i m e { T), o b t a i n e d u s i n g d i l f c r c m s h e l l t h e o r i e s , for a 0/90 c y l i n d e r . - P S T (Tp = 19"8144), -- S D T (I~, .... t 9 ' 8 2 4 '7 i. ..... C S T (Tp = t9.7(}88).
h/a
-- 0,15
-4O -
30
'3 -
20 -10 5
10
15
20
25
h/o = o. 1~
30
35
40
45
/"
(T)
b("
-2O -15 10 -5 5
I0
15
20
25
30
35
40
h/a=O.IS E x / E 0 = z5
&5
(T}
LI~=I ~ . I
/ca : 1/4 q o ~ _ _
F i g . 9. P l o t o f d i s p l a c e m e n t s a n d s t r e s s e s as a f u n c t i o n o f t i m e ('/3, o b t a i n e d u s i n g d i f f e r e n t s h e l l t h e o r i e s , f o r a 0/90 cylinder. - P S T , - . . . . . . S D T , -. . . . . . . CST,
Static and transient response of cylindrical shells
175
I ,3
o.L / ~ -
I
_
"\
",.-X----~
".. I
5
I
I
10
I
l
I
15
~.~---'-~
\
"7"-,.
I
I
20
/
~'~,,/
~
7--" / ' m 7~,
I
I
25
30
I
I
35
I
~
,
",~-'~.~ ,, ---f I
I
.40
I
45
I
(T)
-30 -20 i b x _ 10 -
0 10
,..
\
.
F i g . 1 0 . P l o t o f d i s p l a c e m e n t (~) and stress (~-x) as a function o f time (T), o b t a i n e d for d i f f e r e n t pulse s h a p e s , for a 0/90 cylinder (h/a = 0.1, L / a = 1,/3 = 1. ~la = ~, Exl Eo = 25, T o - 19"814).
higher is the DLF value. However, the DLF value did not exceed 2-0 in the case of ff~, crx and fro. Figures 8 and 9 show the variation of ~ and ~Txwith time (T) obtained using various shell theories. We notice from these figures that the variation of the response curves is periodic with a period almost equal to the longest natural period (values are shown), corresponding to the first flexurai mode of vibration. The irregular or zig-zag nature of these curves indicates the contribution of the higher harmonics associated with the localized nature of the loading. Also, the peak values of ~ predicted by SDT are high when compared with PST, whereas those of CST are lower. The maximum f~ response predicted by CST and SDT is always less than that predicted by PST. Figure 10 shows the response curves obtained using PST for various pulse shapes. The pulse duration time has been taken to be equal to the first natural period (Tp = 19"8144). It is clear that the maximum response occurs during the application of the loading and the response curves for the symmetric triangular impulse and the sine-wave impulse follow similar trends. They are almost identical in the free vibration region ( T > To). The response curves of decreasing and increasing triangular impulses are out of phase with each other in the free vibration region.
176
,4. Bhirnaraddi 5 CONCLUSIONS
In conclusion, we note that the stress and displacement distribution across the thickness of the shell is nonlinear and this nonlinearity increases with the increase in thickness and in the material orthotropy, and with the decrease in the localized load dimensions. The linear stress and displacement distributions of the classical and the shear deformation theories diffel: considerably from the nonlinear distribution predicted by the proposed shell theory. The dynamic load factor can assume values greater thal~ 2.{) depending upon the load geometry.
ACKNOWLEDGEMENTS Financial assistance, in the form of a writing award, was provided by the Faculty of Engineering, University of Melbourne, Australia. In addition~ the author thanks Professor L. K. Stevens and the referee for their c o m m e n t s and suggestions.
REFERENCES l. Donnell, L. H., Stability of Thin Walled Tubes in 7k)rsion, NACA Report No. 479, 1933. 2. FlLigge, W., Stresses in Shells, Springer-Vcrlag, Berlin, 1962. 3. Love, A, E. H., Treatise on the Mathematical Theory of Elasticity, Dover, Nc~ York, 1944. 4. Gol'denveizer, A. L., Theory c~fElastic Thin Shells, Pergamon Press, Oxford, 1961. 5. Sanders, J. L. Jr, An Improved PTrst-Approximation Theory.fi~r Thin Shells. NASA TR R-24, June 1959. 6. Reissner, E. and Tsai, W. T., Pure bending, stretching, and twisting of laminated anisotropic shells, Trans, ASME, J. Appl. Mech., 39 (1972) 148-54~ 7. Bert, C. W., Analysis of Shells, in Structural Design and Analysis, Part t (Chamis, C. C., ed.), Vol. 7 of Composite Materials (Broutman, L. J. and Krock, R. H., eds), Academic Press, New York, 1974, Chapter 5,207-58. 8. Greenberg, J. B. and Stavsky, Y., Vibrations of laminated filament wound cylindrical shells, A I A A J., 19 (1981) 1055-62. 9. Bijlaard, P. P., Stresses from radial loads in cylindrical pressure vessels, The Welding J., 33 (1954) 615s-23s. 10. Vlasov, V. Z., Basic Differential Equations in General Theory o]'Elastic Shells.. NASA TM 1241, 1951. 11. Mirsky, I. and Herrmann, G., Nonaxially symmetric motions of cylindrical shells, J. Acoust. Soc. Am., 29 (1957) 1116-24.
Static and transient response of cylindrical shells
177
12. Naghdi, P. M., On a variational theorem in elasticity and its application to shell theory, Trans. ASME, J. Appl. Mech., 31 (1964)647-53. 13. Zukas, J. A. and Vinson, J. R., Laminated transversely isotropic cylindrical shells, Trans. ASME, J. Appl. Mech., 38 (1971) 400-7. 14. Dong, S. B. and Tso, F. K. W., On a laminated orthotropic shell theory including shear deformation, Trans. ASME, J. Appl. Mech., 39 (1972) 1091-7. 15. Greenberg, J. B. and Stavsky, Y., Vibration of axially compressed laminated orthotropic cylindrical shells, including transverse shear deformation, Acta Mech., 37 (1980) 13-28. 16. Whitney, J. M. and Sun, C. T., A refined theory for laminated anisotropic, cylindrical shells, Trans. ASME, J. Appl. Mech., 41 (1974) 471-6. 17. Gamby, D. and Bourdillon, H., Nonaxially symmetric thermal stresses in thick cylindrical shells, Trans. ASME, J. Appl. Mech., 50 (1983) 217-18. 18. Hildebrand, F. B., Reissner, E. and Thomas, G. B., Notes on the Foundations of the Theory of Small Displacements of Orthotropic Shells, NACA TN 1833, 1949. 19. Naghdi, P. M., On the theory of thin elastic shells, Quartly Appl. Mathe., 14 (1957) 369-80. 21). Miller, R. E. and Boresi, A. P., Strain energy expression for a circular cylindrical shell including transverse shear effects, Proc. 7th Midwestern Mech. Conf., Michigan State Univ., Sept. 6-8 1961, Developments in Mechanics, Vol. 1, J. E. Lay and L. E. Malvern, eds, North-Holland Publishing Co., Amsterdam, 1961,341)-54. 21. Mirsky, I., Vibration of orthotropic, thick, cylindrical shells, J. Acoust. Soc. Am., 36 (1964) 41-51. 22. Nikolai, R. J., Boresi, A. P. and Ford, J. L., A theory for the anisotropic axisymmetric elastic shell with application to the orthotropic cylinder, Proc. 4th Southeastern Conf. Theoretical and Applied Mechanics, Tulane Univ., New Orleans, Louisiana, Feb. 29-Mar. 1 1968. Developments in Theoretical and Applied Mechanics, Vol. 4, Daniel Frederick, ed., Pergamon Press, 19711, 199-220. 23. Golub, E. B. and Romano, F., A method for obtaining stresses and displacements in thick cylindrical shells under arbitrary boundary conditions, Trans. ASME, J. Appl. Mech., 40 (1973) 221-6. 24. Benzley, S. E., Hutchinson, J. R. and Key, S. W., A dynamic shell theory coupling thickness stress wave effects with gross structural response, Trans. ASME, J. Appl. Mech., 40 (1973) 731-5. 25. Lo, K. H., Christensen, R. M. and Wu, E. M., A high-order theory of plate deformation, Pt-l: homogeneous plates, Trans. ASME, J. Appl. Mech., 44 (1977) 663-8. 26. Widera, G. E. O. and Logan, D. L., Refined theories for nonhomogeneous anisotropic cylindrical shells; Part I: derivation, Proc. ASCE, J. Engng Mech. Div., 106 (1980) 1053-74. 27. Logan, D. L. and Widera, G. E. O., Refined theories for nonhomogeneous anisotropic cylindrical shells; Part II: applications, Proc. ASCE, J. Engng Mech. Div., 106 (1980) 1075-90. 28. Levinson, M., An accurate, simple theory of the statics and dynamics of elastic plates, Mech. Res. Commun., 7 (1980) 343-50.
178
A. Bhirnaraddi
29. Bhimaraddi, A. and Stevens, L. K., A higher order theory for free vibration ot orthotropic, homogeneous, and laminated rectangular plates, Tram. ASME~ J. Appl. Mech., 51 (1984) 195-8. 30. Reddy, J. N., A refined nonlinear theory of plates with transverse shear deformation, Int. J. Solids and Structures, 20 (1984) 881-96. 31. Reddy, J. N., A simple higher-order theory for laminated composite piate~, Trans. ASME, J. Appl. Mech., 51 (1984) 745-52. 32. Bhimaraddi, A. and Stevens, L. K., On the higher order theories in piatc~ and shells, Int. J. Struct., 6 (1986) 35-50. 33. Bhimaraddi, A., A higher order theory for free vibration analysis of circtflar cylindrical shells, Int. J. Solids" and Structures, 20 (1984) 623-3(i. 34. Bhimaraddi, A., A higher order theory for dynamic response of orthotropic homogeneous and laminated cylindrical shells, AIAA J., 23 (1985) 1834 7 35. Armenakas, A. E., Gazis, D. and Herrmann, G., Free Vibration of ('itcular Cylindrical Shells, Pergamon Press, Oxford, t969. 36. Bathe, K.-J. and Wilson, E. L., Numerical Methods" in Finite Element Amth:sc~. Prentice-Hall of India Pty Ltd, New Delhi, 1978. 37. Newmark, N. M., A method of computation for structural dynamics, ,,"to, ASCE, J. Engng Mech. Div., 85 (1959) 67-94. 38. Bhimaraddi, A., Static and dynamic response of plates and shells. PhD 'l'hcsls~ Department of Civil Engineering, University of Melbourne, February 1985 39. Timoshenko, S. and Goodier, J. N., Theorv of Elasticity, 2nd edn, McGra~Hill, New York, 1951.
APPENDIX Definitions of 1~ appearing in eqns (8)
,1,4,7] 12 15 /~ 13 16 19
;
[,,o,,, 113,1 1 ,,4 ,,7/ =f 112 I15
(ai
z2
(a+z)~,
, dz aq- Z a ¢-Z
ib
I18.]
119120] 134 135
,]
(I,a + z,$)dz
=
z I (l,~)dz
(a+ z)2I a+t
to)
Static and transient response of cylindrical shells
121= y
(d)
(2(a+z)3dz
(122"123'124) = y
179
( ~ Z ) (Z2'z4'~2)dZ
(e)
126 I~
127 159
=f
z(2a+z)
a
+Z,
1
a+z
dz
(f)
144 142
[ 14° ]
1
= y Z2(2a+z)
/ 30
dz
2a+ z
LV~:.
2,331 , °+~,]
L l2_s 141
(g)
136
~2
(l,a + Z,(a + Z)")dz
(h)
(a + z) 2
(z, z~)dz
[39 [38
(2a + c)
(i)