STATIC ANALYSIS OF MODERATELYTHICK FINITE ANTISYMMETRIC ANGLEPLY CYLINDRICAL PANELS AND SHELLS REAZ A. CHAUDHURI Department
and KAMAL R.
ABUARJA+
of Civil Engineering, University of Utah, 3220 Merrill Salt Lake City, UT 84112. U.S.A. (Received 28 December 1989
: in
revi.red/iwm
Engineering
Building.
I Auyusr 1990)
AbstractThe analytical (exact in the limit) or strong (or differential) form of solutions to the benchmark problems of(i) axisymmetric angleply circular cylindrical panels of rectangular planform and (ii) circumferentially complete circular cylindrical shells. subjected to transverse load and with SS2type simplysupported boundary conditions prescribed at the edges, are presented. The problems investigated. which were hitherto thought to be incapable of admitting analytical solutions, have been solved. utilizing a recently developed novel boundarydiscontinuous double Fourier series approach. for three kinematic relations. which are extensions of those due IO Sanders. Love and Donnell to the firstorder shear deformation theory (FSDT). Numerical results presented for twolayer square antisymmetric angleply panels. which demonstrate good convergence, and show the efl&ts of fiber orientation and thickness on the static response of these panels, should serve as baseline solutions (in the content of FSDT) for future comparison with various approximate weak forms of solutions with either local (e.g. finite element methods) or global supports (e.g. RaleighRitz. Galerkin).
I. INTRODlJCTlON
Analysis of laminated
circular (circumfcrcntially)
complctc cylindrical
shells (c.g. rocket
motor cases, submcrsiblcs. nuclear reactors, prcssurc vcsscls, pipes. tubes, etc.) and circular cylindrical panels (open shallow shells used in aircraft fusclagcs, wings, ships. roofs. etc.) is of current interest because of their increasing use in acrospacc. hydrospuce, energy. chemical and other industrial
applications.
It is a common
shell structures by using such popular ods (FEM).
numerical
because of their inherent complexities
and other coupling
practice to analyze these laminated
techniques as the finite clement mcthintroduced
effects (Seide and Chaudhuri.
by the bendingstretching
1987). Confidence
in these approxi
mate numerical techniques is dependent upon the closeness of agreement of the numerically predicted displacements and stresses of certain benchmark counterparts. cylindrical
Thin
or moderately
problems with their analytical
thick crossply and antisymmetric
shells and panels, with certain types of simplysupported
angleply
circular
boundary conditions,
are two excellent examples of such benchmark problems, which have attracted considerable attention in recent years. Closedform
solutions have been obtained
subjected to axisymmetric and Vinson (1971).
internal
for the case of circular cylindrical
pressure with/without
Reuter (1972). Chaudhuri
temperature
shells
changes by Zukas
et ul. (1986), and AbuArja
and Chaudhuri
(1989) and for axisymmetric buckling of such shells by Hirano (1979). Analytical (e.g. double Fourier series, which are either exact or exact in the limit) solutions have been presented by Stavsky and Lowey (1971). Dong and Tso (1972). Jones and Morgan Sinha and Rath (1976). Tzivanidis
(1982).
Greenberg
and Stavsky (1980).
Hsu et ul. (1981).
Bert and Kumar (1982) and Reddy (1984). for nonaxisymmetric
mation, buckling and vibration of crossply circular (circumferentially) shells and panels, with SS3type [under the classification
(1975),
Soldatos and defor
complete cylindrical
of Hoff and Rehfield (1965) and
Chaudhuri ef al. (1986)] simplysupported boundary conditions prescribed at the edges. Soldatos (1984) has used a Fluggetype theory and Galerkin’s approach in solving the free vibration
problem of noncircular
crossply cylindrical
t Present address: LCC Siporex. P.O. Box 6230. Riyadh
shells.
11442. Saudi Arabia.
R. A. CHAUVHLKI and K. R. AmARJA
2
With regard to the problems of antisymmetric (1983a.b)
and Whitney
angleply cylindrical
(1984) have resorted to Galerkin’s
panels. Soldatos
method. Soldatos (1983b) attri
butes the nonexistence of an exact solution for an angleply cylindrical
panel to the com
plexity of the geometry of a shell (as compared to a flat plate) besides the material properties. by concluding, “thus, for equations of motion of antisymmetric cylindrical
panels and shells whose geometry
other types of exact (i.e. in the limit) or analytical
have been attempted
an exact closedform
While this statement does not attribute impossibility
solution seems to be impossible”. obtaining
angleply laminated circular
is more complicated
of
solutions. no such solutions
by either Soldatos or any other researcher, thus leaving a critical
analytical vacuum. Reddy (1984) has succeeded in obtaining
Navier’s solution in the form
of double Fourier series to the problems of crossply laminated
shells. and has concluded
that “Closedform
solutions for deflections and natural frequencies of simply supported, crossply laminated . . . shells are derived . . . . Unlike plates. antisymmetric angleply laminated shells with simplysupported boundary conditions do not admit exact solutions. The exact solutions presented herein for cylindrical and spherical crossply shells under sinusoidal. uniformly for approximate
distributed,
and point loads should serve as benchmark
methods, such as finite element and finite difference methods”.
results
It may be
pointed out here that there appears to be some confusion or controversy with regard to the definition
of “exact solution”
natural frequencies. obtained
in the literature.
While
expressions for eigcnvalues.
by both Reddy (1984) and Soldatos and Tzivanidis
e.g.
(I‘%?)
for crossply shells are in closed form, the solutions for dcflcction prcscnted by the former are dctinitely not in closed form. Rcddy’s (1984) solutions for dctlcction of crossply shells can only he rcgardcd as solutions of strong (or dilTcrcnti:ll) form. or solutions exact in the limit or exact in the scnsc of Chia (IWO. p. 3X) stated. “The solution
and
equations can bc truncated
S/il;trd ( 107l. p. 43). Chia (I9SO) has
SCIISC th:~t ;III
is said to hc oxact in the
. . ;llgchraic
infinite set of
to obtain any dcsircd dcgrcc of accuracy. Enact solutions will
bc obtnincd by douhlc Fourier sorics. gencrulizctl douhlc Fourier scrics and a combination of thcsc scrics . . .“. According
to Szilard
( 1974). “In
‘exact’ solutions . . . : I. Closedform
mathciiiatically
gcncral. thcrc arc four types ol
solution. 2 . 3. Double trigonometric
scrics solution. 4. Single scrics solution”. Recently. Chnudhuri
(1989) has presented iI novel msthod for obtaining an analytical
[exact in the limit or exact in the sense
of solution,
of Chia
(IWO) and Srilard (1974)) or strong form
in the form of boundarydiscontinuous
of a system of completely
coupled
linear
double Fourier series, to the problem
partial
differential
equations
with
constant
coefficients, subjected to completely coupled boundary conditions. This method (i) provides guidance with regard to the selection of appropriate on the coetlicients
assumed double Fourier series solution
functions,
depending
equations,
(ii) guides in making decisions with regard to the discontinuities
of the system of governing
assumed solution functions or their first derivatives, boundary
condition
equations.
(or highly) coupled PDEs. Chaudhuri
(1987.
(iv)
1989) has also applied this method to invesFourier
series solutions
in the case of
laminated anisotropic shells of rectangular planform.
lytical solutions to the problems cumferentially
and finally.
scheme in spite of the complexity of the completely
tigate the general nature of these exact double doublycurved
difTcrential either in the
depending on the coctlicients of the
(iii) ensures uniqueness of the solution,
leads to a highly efficient computational
moderatelythick
partial
of antisymmetric
anglcply
cylindrical
Ana
panels and cir
complete cylindrical shells have, however. not been investigated in detail and
their numerical results are still nonexistent. The primary objective of the present study is to apply the aforementioned in obtaining a unique solution to the five highly coupled secondorder
technique
PDEs subjcctcd to
highly coupled boundary conditions. This study will present (i) analytical
(double Fourier
series. which are exact in the limit) or strong (or differential) forms of solutions to the aforementioned benchmark problems for three kinematic relations (shell theories). which are extensions, by Bert and Kumar (1982). of those due to Donnell, Love and Sanders to the cast of firstorder
shear deformation
results for antisymmetric
theory (FSDT)
and (ii) some useful numerical
angleply cylindrical panels limited to Sanders’ kinematic relations
alone. as a first step. Although
the problem of doublycurved
anglcply panels was solved
Static analysis of cylindrical
panels and shells
3
by Chaudhuri and AbuArja (1988) using a preliminary version (Chaudhuri. 1987) of the boundarydiscontinuous double Fourier approach due to Chaudhuri (1989). the criteria determining when the boundary Fourier series are needed or not needed were neither fully understood, in the mathematical sense. by the authors at that time, nor were ever fully investigated by such previous investigators as Goldstein (1936. 1937). Green (1944). Green and Hearmon (1945). Whitney (1970. 1971) and Whitney and Leissa (1970). This issue has now been completely investigated. in the mathematical sense. by Chaudhuri (1989). The present analysis on antisymmetric angleply cylindrical panels and shells is, therefore. firmly rooted to the mathematical analysis. while keeping the door open to physical interpretation by the intuitive approach of the aforementioned earlier investigations. The numerical results presented here are expected to serve as benchmark solutions (in the context of FSDT) for future comparison with various approximate weak (or integral) forms of solutions with either local (e.g. finite element methods) or global supports (RaleighRitz. Galerkin). Definitions of strong and weak forms are available in Hughes (1987). The scope of the present study will be limited to the type of prescribed boundary conditions (i.e. SS2) considered by Soldatos (1983b) and Reddy (1984) for antisymmetric angleply panels. although the remaining boundary conditions can be handled with almost equal ease.
2. STATEMENT
OF
THE
PROBLEM
For a cylindrical panel shown in Fig. 1. with axial and circumferential lengths. o and h. rcspcctivcly, and radius, R. the straindisplaccment relations, under FSDT (ReissnerMindlin hypothesis) arc
whcrc
in which a comma denotes partial ditl’crentiation. E,and E: (i = I, 2.6) represent the surfaceparallel normal and shearing strain components at a parallel surface and midsurface respcctivcly. while E,. .$’ (i = 4.5) represent the corresponding transverse shearing strain components. K, (i = I, 2.6) denotes the changes of curvature and twist, while U,(i = 1.2.3) and 4, (i = 1.2) denote the displacement and rotation respectively in the ith direction. The cocllicients ((.,.c?) are shell theory tracers, for the extensions of Sanders’. Love’s and Donncll’s theories respectively to their FSDT counterparts (Bert and Kumar, 1982). The equations of equilibrium are :
Fig.
I. A
circular cylindrical
panel.
R. A. CHAUDHL’RI and K. R. ABUAIUA
N,,, +Nc C’2M6.2/R= 0: NdRQt.,
Qz.zq
= 0;
N,, + FZM6., lR + N2,2 + C,QJR = 0 ;
M,,,+M,.2Q,
= 0;
M,.,+IL~:.~Q~
=0
(3)
in which q is the transverse or radial distributed load. Surfaceparallel stress resultants, N,. stress couples (moment resultants), Mi, and transverse shear stress resultants, Q,. are related to the midsurface strains. ~9, and changes of curvature and twist. K,, by N,=A,,&,O+B,,x,
(i,j=
1,2.6);
Q, = A&+4&
Mi=Bli~~+D,,~,
(i,j=
1,2,6);
Qz = AJ.g:+~&
(4)
Here A,,. B,,. D,j are extensional, coupling. and bending rigidities, respectively, and A,, (i. j = 4.5) represents transverse shear rigidities. For an antisymmetric angleply laminate, A,6
= AZ6 =
Ad5 = B,, = B12 = B,, = Bb6 = D,, = D:b = 0.
(5)
Substitution of eqns (2). (4). (5) into eqns (3) will yield five coupled partial differential equations with constant coefficients in the following form :
whcrc the supsrscripts of the coctiicicnts. i. denote the equation number. u:” (i = I,. ,5; j= 1..... 18) arc as dcfincd in Appendix B. The five boundary conditions at an cdgc arc sclcctcd to bs one member from each pair of the following:
(% Nn)= (I(,.1,) = (10,Q,,) = (4”. M,) = (4,. M,) = 0 at an edge .v, = 0.
(7)
The boundary conditions considered here are the same as those considered by Soldatos (1983b) and Reddy (1984). which are termed SS2type under the classification of Hoff and Rehfield ( 1965) and Chaudhuri Edal. (1986). They are prescribed xs follows : UI (O.r:) =
NP,(O..TZ) = ~.t(O,.rz) = M,(O.,Y~) = &?(O,x2) = 0 at the edge _y, = 0. (8) 5. SOLUTION
FOR CYLINDRICAL
PANELS
It has been shown by Chaudhuri (1989) that selection of the assumed boundarydiscontinuous double Fourier series solution functions will depend on the governing partial differential equations and not the boundary conditions. At stake hcrc is the wellposedncss of the Fourier analysis. to be achieved through selection of the unknown coefficients of the assumed double Fourier series solution functions and introduction of certain boundarydiscontinuous coefficients, so that the number of final algebraic equations become equal to the number of unknowns to furnish a unique complete solution. The details are available in Chaudhuri (1989) and will. in the interest of brevity, be excluded here. The solution to the system of five coupled partial differential equations, given by eqns (6) in conjunction with the SSZtype simplysupported boundary conditions. represented by eqns (8). will then be assumed in the form :
St;ttic an;ti>sis olcyfindrical pan&s and sficils where
in which
The above equations
( lOtttn f Star + Sn + I) unknown
introduce
The next opcrrttion will comprise dit~er~nti~tion is a necessary step before substitution
Fourier
coefficients.
of the assumed solution functions, which
into the equilibrium
eqns (6). The procedure
for
dil~~r~nti~tion of the assumed double Fourier series solution functions for the most general type of boundary condition
has been described in detail by Chaudhuri
c&ion of which to the prcscnt problem is rcl&ivcly proccdurc,
described in Appendix
st~~i~htfo~~rd.
(1989).
the appli
The illustration
of the
A, will. in the interest of brevity of presentation,
bo
limited to obtaining the first prtrtinl dcrivativcs of the assumed solution function, u:‘. These twod~riv;~t~v~s directly follow from eyns (A?bf. (A&).
(A8) and (Al i) and can be rewritten,
in the final form:
where
for i odd for i even
03)
and
4 * c, = f u:‘
(I&)
(l4b)
Extension of the above procedure to the second derivatives is straightforward 1989) and will yield :
(Ch~udhuri.
6
R. A. CHNDHURI
and K. R. AmARJA
x {Bxc!n+Bnhnj.n+&S,);
( I5c)
cos (z,.‘c,) sin (&x~)
where
{u:!,(a,.r:)u’,!,(O,.rZ))
sin(P,.rz)d.rz
(16a)
A similar procedure applied to the other assumed functions leads to I6 more constants: en,,
/I,. i., jn, HI,, n,, qn, r,. for each n, the four pairs being associated with uy, &,, respectively along the boundaries xl = 0,~; and g,,,, h,. k,,,. I,,,, o,,, P,,,. s,. I,. the four pairs being associated with u$, .v? = 0.h.
a!;.:, #,!,.
&‘, respectively along the boundaries
It may bc noted that the first and second dcrivativcs
assumed solution functionsuj ditfcrcntiation presentation.
(Chaudhuri.
(i = I. 2.3) and 4,’ (i = I, 2)can
1989). Thcdctails
#,‘. (by.,.
for each nr,
will bcomitted
of the first parts of the bc obtained by tcrmwisc
here in the interest of brevity of
The above step introduces (IOVZ+ IOn+ 12) additional
unknown coetficicnts,
which ask for as many equations, to be supplied by the boundary conditions to bc presented later. It will be interesting to offer the following
physical explanation
of the above mathc
matical operations, as applied to the present problem : The first parts of the assumed solution functions 111(i = I, 2.3) and (bf (i = I, 2) which are the same as those assumed by Reddy (1984) boundary conditions, conditions
as
complete solutions, satisfy the prescribed
given by eqns (S), u priori. Since they do not violate any physical
at the boundary,
their partial derivatives can be obtained
by termwise differ
entiation. The second parts by themselves do not satisfy the prescribed boundary conditions. Hence, according to Chaudhuri’s boundary
conditions
of other physical conditions derivative(s).
Chaudhuri
at certain
(1945).
may arise out of violation
by these functions
and/or
their first
of the assumed functions and/or their
concerned may not be valid, as stipulated by Green (1944), Green Winslow
(1951)
(1989). The corresponding
by expanding
boundaries
In such events, termwise differentiation
first partial derivative(s) and Hearmon
(1989) method, they are forced to satisfy the prescribed
given by eqns (8). Still, discontinuities
Whitney
and Leissa (1969).
Whitney
first or second partial derivative(s)
them in the form of double Fourier
(1971)
and
must be obtained
series, in a manner suggested by the
above investigators. This will be illustrated for the case of the two first partial dcrivativcs of the assumed solution function, u:‘, which does not automatically satisfy the prescribed geometric
boundary
conditions
at the edges x, = 0,~;
these are then satisfied by force.
which will yield additional equations arising out of satisfying boundary conditions to be described later. u’,’ however, vanishes at the edges x1 = 0, h, which is a violation, because at these edges u, = u, and N, = N, cannot, according to the variational principle, bc simultaneously prescribed. Therefore, u‘,!2 cannot be obtained by termwise differentiation, although u’,‘., can be obtained that way. Expansion of the uniformly
distributed
transverse load into a double Fourier series
4 = 1 1 qmnsin (cx,x,) sin (/3,x2) ml “I
(17)
Static analysis of cylindrical
and substitution
7
panels and shells
of the assumed solution functions and their appropriate partial derivatives
into eqns (6a) will yield :
i
sin (r,,s,)
(((I’;’ r,;,u(?‘)U!
,111+ Y!“,,(N’,? a,$;:)
 ia,,,C,,cl’i
+ V!$,(tr’{’ aill\:‘)
ml


1 2a,,,c,,~,,,tr~~’  ia ,,,,/;,S,,,u’,:’+ &j,,,tr:”r,,, CV!/,,u!;’ + .Yj,l,,(u’/I z&‘;~)
:a,,,m,,y,,,u’i~  ia,,,tlUij,,d;~
+ ~Cl,,,U’i\  ~N’i~Z ,,,.V,,] = 0
(18~)
where the superscript i takes on the values i = I ,3,4 and 5. Similar of eqn (I 7) will yield :
operation on eqn (6b)
,L,
fln?rry’)
and substitution
.,$, sin (r ,,,.rg,) sin (/~,,.Y~){  /I,,tr\“Uk, a,,,d~‘X!,,
/Id,!
Yf,,, a,,,rl’,“U~,
Vf ,,,, + (u:” 
a,&”
/I,,u\”
VI, +a,,,/I,u:”
r~,a~”
Wk,
bfl!,
mg, “$,
COS(x,“.Y,)COS(~~,.\.~)(~,,,O’,“L/~,+/I,N:1’v/a,+r,,,~,ll’~’Cvj,,+~.cl’,:‘x~“,
+ a&,:’
YL, + flnui”
I!/~!. + df3’c,;‘,
+ (N’g”  z,f,rr:,”  /?I&‘) +a,,c7:“X!!, m$, cos (ZJ,)~
+u!f’m,y,, ~,,,a’,” cl;,
+ (0:”  c&r’,“)
+ z,dJ3’ v;;, + a’,“‘e,y, + U:l’f”5,
CV!,‘,,+ i.ymuk” +j,S,a~“’ +u:?‘n,dv,
+/I.u,,~
+ !j”d,uC,”
+ $l’,~.T,,#; = 0
+ k,y,ul;”
+ I,S,a’,“’
UC, +a,,‘z’s,y,+fl’,~f,6,)
+ z,,u’,:) Yb, + !c,uy’
Wt:,, + ii,y,,tlb!’
+ py”f?t ,,;‘“, + iu!:‘ll,)b,.,
+ a\“[email protected],
+ amu\” Vi,
= 0
+ ia:“eoyn
(19b)
+ $~\‘)/bd,
+ {t~‘,~‘k, +z,~:“X~~ (19c)
R. A. CHALDHLRIand K. R. AeuAIUA
!!
(Usl’ /gu:?‘) Lb:;”+ &&l’i” + {Ub”i’“k,, + ~u’,“&f,, + ~u:J’m,+
+
+ ~tr’,~.s,,;~, + ~tr’,I’t,,&) = :Ll;“(.(, + :u’,“e,, + Equating contribute
uy Cl”’,,,,+
)!?.u’,‘I’ Yl:,
0
(19d)
&lpi” + fU’yJ’k”+ 4Y ~u’J’M0 + 4la”‘3 I? [email protected]
(IYe)
the coefficients of the trigonometric
lOnzn+Sm+5n+
I linear algebraic
functions
equations.
of eqns (18) and (19) will
The remaining
IOm+ IOn+ I2
linear algebraic equations are supplied by the geometric and natural boundary conditions, given
by eqns (8).
For example,
satisfaction
of the geometric
boundary
conditions
II, (0. s:) = II, ((1, .v:) = 0 and equating the coefficients of sin (lJnxz) will contribute
the fol
lowing 31 linear algebraic equations :
,,& ,5,”CC!”= 0; The remaining
lOr~+8/1+
CJ::, + i
,,,z I
=o:
forall
I2 equations are contributed
conditions of crlns (8). Therefore. of IOMU+ I51rr+ I5nf
y,u!!,
t1=
(20)
1,2....
by the rest of the boundary
dcpcnding on the dcsircd degree of accuracy, a finite set
I3 linear algebraic
solved. In the intcrcst ofconiput;itional
equations
in as many unknowns
needs to be
cfticicncy, cqns (18). (19) arc solved for U:,,,. V:,,,.
cl’:I,,,. .\.:,,,,, }Z,,,. i = I, II in terms of the constilnt coctiicicnts u,. h,, cn,, tf,“, etc. which arc then suhstitutctl into the cqu;ltions ilrising out of the boundary conditions [e.g. cqns (I?())] lin;illy yielding (I,,. h,,. (‘,,,, cl,,,, etc. following ;ln approach suggested by Chaudhuri (IWW). This operation will rcducc the si/c of the problem under consideration by iIn order 01 m;igniludc, unknowns.
limilly
resulting
in IOM+
IOff + I 2 linear
It m;iy bc noted that for an xntisymmctric
Substitution Furthermore,
algebraic
equations
in iIs many
the solutions of which ;lrc rol;ltivcly speaking triviill. angleply plate (K = c5),
of cqn (21) into eqns (6) will reduce them to their flat plate counterparts. substitution ofeqn (2 I) into the final solutions [eqns (IS)(20)]
to yield the corresponding (1984). Substitution
can be shown
Rat plate solution presented by Bert and Chen (1978) and Reddy
of
will rcducc cqns (6) to their counterparts
for a homogeneous orthotropic
plateeqns
(8).
(A I) of Chaudhuri and Kabir (1989). who have presented solutions only for the SSJtype simplysupported amI C4type clamped boundary conditions. However, solution in the form of linear algchraic equations for the SS4type boundary conditions can easily be modified to obtain their SS2type counterparts. which can be shown to be identical to those obtained by substitution ofcqn (22) into eqns (IS)(20). thedetails ofwhich will be omitted here in the interest of brevity. This will be further dealt with in Section 5.
Stilt~c analysis of cylindrical panels and shells 1. SOLUTION
FOR A CIRCUMFERENTIALLY
For a circumferentially
9
COMPLETE CYLINDRICAL SHELL
complete circular cylindrical shell. the x2 coordinate
is replaced
0. where .‘c: = R8. The resulting governing partial differential
by the angular coordinate.
equations are similar to eqns (6). The assumed solution functions are still given by eqns (9). (IO) with the argument /?,,x~ replaced by no. where n is as before, an integer. Since the assumed trigonometric circumferentially
functions
complete
of no automatically
cylindrical
satisfy the closure condition
shell. the partial
derivatives
solution functions with respect to 8 can be obtained by termwise differentiation. assumed solution functions with superscript II. and/or respect to I,
may have edge discontinuities,
of a
of all the assumed Only the
their first partial derivatives with
in which case termwise differentiation
will no
longer be valid. This will be illustrated for the partial derivatives II’,‘,, and u’,‘,,, as before :
u’~‘.,(.u~.O) = 
u:‘.,,(s,.o,
= ; i
(0 < X, < a)
f i U~,lx,sin(~,.r,)sin(nO); In I “= I
a,sin(nO)+
2
m= I
f
(z~C/!J,+
a,g,, +M,)cos
(23a)
(z,,.r,) sin (no).
m= I n= I
(23b) Comparison
with the preceding section will reveal that all the unknown
with subscript ~1, e.g. cm. cl,,,. will vanish in the case of a circumfcrentially cylindrical
shell. Following
coefficients
complctc circular
an identical procedure as in the preceding section, a finite set
of I Onrrr+ 5~ + 5n+ 7 equations in terms of the identical number of unknowns is generated. Following
the proccdurc
given by Chaudhuri
(1989).
thcsc are reduced to a system of
5m+ 5rr + 6 linear algebraic equations in as many unknowns, u,,. h,. . . . , which can be easily so I vcd .
5. NUMEKICAL
The present study will investigate, moderatelythick
cylindrical
RESULTS :ISan
AND
DISCUSSIONS
example, a twolayer antisymmetric
panel of square planform
angleply
(u = h) with fiber orientation
of
g/jrT. The layers are of equal thickness. The length, u, and the mean radius, R, of the cylindrical
panel are 8 12.8 mm (32 in.) and 2438 mm (96 in.), respectively. The uniformly
distributed
pressure, (1, is 0.6895 GPa (100 ksi). E,( = EL) and E?( = Er), Young’s moduli
of a layer in the directions parallel and transverse to the fiber direction, are 172.4 and 6.895 GPa (25.000 and 1000 ksi). respectively. The shear modulus GZ3 (= C,)
is assumed to be
equal to 3.448 GPa (500 ksi). The shear moduli. G,: and G,, ( =GL.r). are assumed to be the same in the case of highly crystalline graphite fiberreinforced
composites. The major
Poisson’s ratio, V, I (= vLr), is take n equal to 0.25. Sanders’ kinematic to FSDT.
have been used in the computation.
solutions for the corresponding metric parameters following
spherical panels of otherwise
have recently become available
nondimensionalized
(Chaudhuri
identical material and AbuArja,
and geo1988). The
quantities arc defined :
14:= 320E~h~u,/(yn’); U? = 320&lz’u&~‘)
Al:=
relations, extended
This example is selected, because analytical
1024Mi/(yu’);
i = 1,2;
; i=
IIT+ = IOU?
c#Jr= lOc#): 1.2.
(24)
Table I presents a convergence study of UT, UT. (6,. h4: for various aspect ratios. U/IL and fiber orientation angles, CT.14:and My are computed at the center of the panel, while u: and 4, arc computed at the midpoints of the two sides, (u/2.0) and (O./J/~). respectively. It may be noted that not only the sidetothickness ratio, o/h, but also 6 plays a role in the convergcncc of displacements. rotations and bending moments. Convergence rates of ur.
R. A. CHAGDHCRI and K.
IO Tublr
R. ABCARJA
I. Convergence of UT (at the center), u: (at .x, = (I 2. xz = 0). 4, (at x, = 0. .x2 = h’2). and .V: (at the cmtrr) for VXIOUS uspcxt ratios. CJh.andfiber onentatwn angle. ir
0 uh
IO
20 IO 20
Ii:
20 IO 20
.\I:
4:. My are similar
3.505 2.441
3.547 2.4YJ
3.548 1.4’26
I.754 1.357
1.762 1.367
2.979 19.05
3.OOY I9.44
3.OlY 19.57
79.63 6’.‘5 . .
XI.78 64.59
RI.23 64.30
2.737 I .Y24
0
0
0
I.725
0
0
0
I .?YY
I.932 13.98
I.932 13.99
125.2 111.3
to those of UT. 4, and M:.
114.9 II?.0
respectively, and hence these are not shown
here. The convergences for the displacements and rotations The results
8
2.738 I .Y’?
123.3 110.0
indrical panels shown in Table
6
2.708 1.xY6
I.YII 13.75
IO
$4
4
8
6
4
m = n
u:
22.5
0
of two moderatelythick
cyl
I are reasonably rapid and may be regarded monotonic.
for the special case of homogeneous orthotropic
shells ((7 = 0 ) compare fav
orably well with those obtained using an cxtcnsion of the approach presented by Chaudhuri and Kabir (198’)) to the C;ISCof SEtype
boundary conditions.
These results testify to the
validity of the analytical approach and the accuracy of the numerical results. noted that the convergence for ths moment. MT. acccptabloaccording to the theory of Fourier
exhibits
series, as expounded by tiobson
amplitude
of oscillation,
the rwan
vaiuc incrcascs. and the amplitude of oscillation
that arc prescntcd Mow
scrvcs ;IS 1111 error norm. This
and Kabir
along the crntcrline.
point of iI side parallel to the .r+xis.
d,, at the mid
(0. h/2). The variation of the central moments with
angle, 6, is presented in Fig.
I
of ;I side parallel to the _v,
displacement, UT, and the rotation,
5. It may be noted that the transverse
displncemcnt, ti’, and the bending moment, icf ,. are symmetric
__
..
u,
with respect to the centerline
*I
3 1 Flp. 2. Vari;lrion
panel
angle. 0,
transverse displacement, II :. at the center ((r/2,6/2) ; the surface
and the surfaceparallel
tiber orientation
cylindricul
the variation with the tibcr orientation
pitrallcl displaccmcnt, II:. and the rotation, &z, at the midpoint axis. (trj2.0);
results
displacements. rotations
_r: = h/2. of ;I modcratclythick
(U//I = IO) with Cl= 45 ‘. Figure 4 exhibits of the nondimcnsionalizcd
( 19X9). Numcricnl
have been obtained by using 111= N = 8.
f:igures 2 :IIICI 3 show the variation ofthc nondimcnsionnlizcli mumcnts
which is
(1Y26). The
howcvcr, dccrcascs with the incrcasc of the number of terms, while
phcnomcnon has also been obsorvcd by Ch:ludhuri
;IIKI
It may be
bounded oscillation.
of displxcmen(s
and roUtions along the center pUlc1.
hnc. .I! = h;?. of 3 cylindrul
Static analysis of cylindrical
panels and shells
II
x,/a Fig. 3. Variation
of bending moments along the center line. x2 = bi?. of a cylindrtcal panel.
.’
0'
1
15’
30’
45’
i Fig. 4. Variation
ofdisplaccmcnts
and rotations
with the fiber orientation
angle, 0.
i Fig. 5. Variation
of bending moments with the libcr orientation
an&,
0,
= up. while the surfaceparallel displacements and rotations are antisymmetric with respect to the same. Similarity of these plots to their independently computed spherical [see Figs 2S. Chaudhuri and AbuArja (1988)] counterparts testifies to the accuracy of both the sets of results. Currently unavailable numerical results, on (i) comparison with other shell theories (e.g. Donnell and Love), (ii) comparison of free vibration results with those of Soldatos (1987) and Kabir and Chaudhuri (1991). and (iii) circumferentially complete cylindrical shells. will be published in forthcoming papers. A,
12
R. A. CHAUDHUU
and K.
R. AKANA
6. COSCLUSIONS
The analytical the problems
(exact
in the limit)
of moderatelythick
circumferentially
complete
developed
boundarydiscontinuous
novel
facilitates
circular
the wellposedness
coefficients
of the assumed
boundarydiscontinuous equal to the number mark problems be incapable
double
of unknowns
techniques
as the finite element
to the firstorder presented utilizing
for
series solution
to furnish
methods.
version
of Sanders’
Extension
of boundary
(ii) the problem
constraints,
shells, and currently
unavailable
thcorics
(c.g. Donncll
and Love),
complete
cylindrical
presented
of free vibration
numerical
results.
(ii) for free vibration
shells, will be published
(in
numerical here for
Love and Donnell
results.
panels
that
have been
of square
demonstrate
ratio and fiber orientation
of the approach
in the field to
by such popular
Numerical relations.
of
become
The two bench
have been obtained
cylindrical
kinematic
gence, and also the etfects of sidetothickness response of these pan&.
solutions
(FSDT).
angleply
equations
to serve as baselines
of those due to Sanders.
theory
antisymmetric
the extended
Analytical
method
and introduction
by the researchers
with those obtained
a recently
This
of the unknown
solution.
which are expected
which are extensions
shear deformation
twolayer
functions
complete
considered
solutions,
selection
to
panels and
utilizing
of final algebraic
a unique
of solutions
cylindrical
series approach.
through
so that the number
for future comparison
relations,
Fourier
analysis
here were hitherto
of FSDT)
forms
circular
shells have been obtained
Fourier
analytical
the context
(or differential) angleply
double
of the Fourier
investigated
three kinematic
cylindrical
coefficients,
of admitting
or strong
antisymmetric
planform
good conver
angle on the static
herein to (i) include
the eRcct
and (iii) other tvpcs of laminated
on (i) comparison problems
in forthcoming
with
other
shell
and (iii) circunlf~rcnti~llly
papers.
REFERENCES AhuArj;l.
K. R. and C’h;~utlhurl. R. A. (IYXY). Modcralcly
thick cyhndrical shells under inlcrn;ll prcssurr. A.S,I/I:’
J. App1. ~tlcch. 3, 652 657. Bcrr. C’. W. and Chcn. T. L. C. (iY78). Eli&t of shear dcformarlon on vitxlfion of ;tnrl>ymmctric ;~nglcply I;~minalcd reclanyul;lr plates. ftlr. J. Sditls Sm~rurr.s 14. 465 173. Ilcrt. C. W. and Kumar. M. (19X2).Vibrations of cylindrical shells of bimodulur compc)s~~c malcrialr. J. Sound I’rlw. Xl. 107~121. C’haudhuri. R. A. (IYX7). A novel double Four&r series approxh for solutions IO boundary value prohlcms of modcrarclythick laminated shells (abslracl publishrd). Presented a1 thr 24th Annual hlccting of rhr Society of Englncering Sciences. held at Univrrsity of Utah. Salt Lake City. Chaudhuri, R. A. (IYXY). On boundarydiscontinuous double Fourier srrics solution to a nybtcm of complelcly coupled PDEs. /III. J. Engng Sci. 27, 10051022. Chaudhuri. R. A. and AbuArja. K. R. (IYXX). Exuc~ solution of sheartlcxiblc doublycurved ;tn~isymmrtric ;tnyleply shells. brr. 1. Engng .Sci. 26. 587604. Chaudhuri. R. A., Balaraman. K. and Kunukkasseril. V. X. (1986). Arbirrarily laminated anisolropic cylindrical shells under internal pressure. AlAA JI 24, I851 18%. Chaudhuri. R. A. and Kabir. H. R. H. (1989). On analytical solu[ions IO boundaryvalue problems of douhlycurved moderatelythick orthotropic shells. Inr. J. Engng Sci. 27. 1325l 336. Chia, C. Y. (IYXO). Nonlincwr Analysis of Plures. McGrawHill. New York. Dong. S. 8. and Tso. F. K. W. (1972). On a laminated ortholropic shell theory including 1ransvcrsc shear deformation. ASblE 1. Appl. Mech. 39. IOYI1097. Goldkn. S. (IY36). The srabihly of viscous tluid flow under pressure bctwccn p;lr;tllcl plants. Prrw. Cwtrh. /‘h/1. sm.. 32. JO 11). Gold&n. S. (lY37). The slnbili~y of viscous tluid flow belwccn roraling cylinders. Pror. Cumh. Phil. SIW. 33, 41
55 Green. A. E. (194I). Double Fourier series and boundary value problems. Proc. Cumh. Phd. SK. JO, 121 728. Green. A. E. and llcarmon. R. F. S. (1945). The buckling of flat roxangular plywood pIarcs. Phrl. M~tg~irrc* 36. 659 6X7. Grccnbcrg. J. B. and Slavsky. Y. (1980). Buckling and vibration of orlhotropic composi(c cylindrical shells. .4c’/o ,\/c~chunicu 36. 15B. llirano. Y. (lY7Y). Buckling of angleply laminated circular cylindrical shells. AS.\/E J. Appl. .Mc*ch. 46. 223 234. t lobson. E. W. ( I Y26). The 7’ht*ory o/ Funcrion.~ of a Rrul Vuriohlc und the Thcw.v o/‘Fmtriur St*rie. Vol. II. ?nd cdn. Cambridge University Press. U.K. tlotr. N. 1. and Rehfield. L. W. (IY65). Buckling of axially compressed circular cylindrical shells al stresses smaller than the classical critical value. A.S,M& J. Appl. Mcch. 32. 541546. lisu. Y. S.. Rcddy. J. N. and Bcrc. C. W. (1981). Thcrmoclaslicity of circular cylindrical rhclls luminalcd of blmodulus composite materials. 1. T/term. .S~rc*s.~rs 1. 155l 77.
Static analysis of cylindrical
panels and shells
13
Hughes. T. J R. ( 1987). The Fimrr Ekmtwr .Wuthod. PrenticeHail. Englewood Cliffs. New Jersey. Jones. R. XI. and !Uorgan. H. S. ( 19751. Buckling and vibration of crossply laminated circular cylindrical shells. .41.4.4 II 13, 664 671. Kdhlr. H. R H. .md Chaudhurl. R. A. ( I9911. Free vibration of shearflexible antisymmetric angleply doubly cur\ed pan&.
Inr. 1. Sditb
Smccrurc~
28, l732.
Rrddp. J. N. ( IYS1). Enact solutions of moderately thick laminated shells. J. Engng Mech. Dir. ASCE 110. 794 SOY. Rsurrr. R. C. I 1972). Analysis of shells under internal pressure. 1. Cornpus. MuIer. 6.94l 13. S&Jr. P. and Chaudhuri. R. A. (1987). Triangular finite element for analysis of thick laminated shells. hr. 1. .V~tmrr. .\/crh. Ent./n,q 24. 1563l 579 SlnhJ. P. K. llnd Rath. A. K. (1976). Transverse bending of crossply laminated circular cylindrical plates. J. I/~YA. Enyrtu 5c.i. 18. 53 56. Soldatos. K. P. (lY83a). On the buckhng and vibration of antisymmetric angleply laminated circular cylindrical shells. Inr. J. Engn,g Sci. 21. 217212. Soldatos. K. P. ( 1993b). Free vibrations of antisymmetric angleply laminated circular cylindrical panels. Q. J. .Mtz,h. Appl. .Vurh. 36. 207?,I. Soldatos. K. P. (IYYJ). A Fluggetype theory for the analysis of anisotropic laminated noncircular cylindrical shells. Inr. 1. Solitls SIIIICIW~.~ 20. lO7120. Soldatos. K. P. ( 1987). Influence of thickness shear deformation on free vibrations of rectangular plates, cylindrical panels and cylinders of antisymmetric angleply construction. /. Sound V&r. 119. I I l137. Soldatos. K. P. and Tzivanidis. G. J. (1982). Buckling and vibration of crossply laminated circular cylindrical panels. J. .+pl. Murh. Php. (ZAMP) 33, [email protected] SLI\A). 1’. and Lowey. R. (I971 ). On vibrations of heterogeneous orthotropic cyhndrlcal shells. J. Sow~tl bhr.
IZ,‘? ‘3 R. ( IY7J). T/wrrt, turd .Antr/n+s n/ P/trfc*s. PrenticeHall. Englewood Whltnc!. J. xl. ( 1’)70).The cltcct of hound;try conditions on the response Sfll;lrd.
Cliffs.
New Jersey.
of laminated composites. J. Cotupm.
.I/cr/t7 4. I’,? 203. H’hltncy. J kl. (1‘471). Fourier analysis ofclampcd anisotropic plates. ASME 1. Appl. Mach. 1. 530.532. \Vlulncy. J. hl. (IYXJ). Buckling ofanisotropic Iaminatcd cylindrical plates. AIAA I/ 22. 1641 1645. IVhitncy. J. 51. ;~ntl LcI\~;I. A. W. (lY70). An;tlycis of heterogeneous anisotropic plates. AS,ME 1. .App/.
Mcch.
36. 20 I Xh. W~c~slow, A. hl. (IY5l
). I)llfcrcnti;ltion
..I/‘/‘/. .\/~r/h. 4. JJ9 JhO. %uh;~s. J. A ;~nd Vinson. J. R. (lY71). 38. JIM 407.
of Ft)uricr L;ltnin;ttcd
scrics in sfrcss solutions transversely
APPENDIX
for rectangular plates. Q. 1.
isotropic cylindrical
shells. ASME
1. Appl.
Mcch. Alcch.
A
lisrcnlial principles of d~lliirentiation have bcrn expounded by tlohson in his classic volume (1926) on the \uhjcct and revtcwed by Winslow (lY51). in the context of ordinary Fourier series. Extension of IIW same to the c;~seofdouhle Fourier series has been treated by Chaudhuri (IYXY). In grncral, the scricr ohtaincd by dilTcrcnUing a convergent double Fourier series, representing a function F( v,, v,). is not convcrgsnt : neither is the series so obtained necessarily the Fourier series corresponding IO the p;lrtlcular parll:d derivative of F(.r,..r,). Let F(x,.s,) bc a bounded function and be picccwise continuous (i.e. conlinuous cxccpt for a tinite number ordinarydiscontinuities) ; let it also be assumed that the partial derivatives. p,(.\,..v,). I = I.2 havr a Lchcsguc integral in the domuin (U.U) x (h,h) and that if it has lines of inlinite dI\continuity (c.g. Dirac Dsl~n function), such lines form a reducible set. This is consistent with there being a set oflinch of/cro measure at which F,(.v,,,s,) h;ls no dctinite value. We conbldcr as an example u’~(.v,.s~). which. as dclined in cqns (IO). is an even function with rcspcct IO I, ;md an odd function with respect IO .v>.
of
U’:(x,,  .I’?) = u:‘(s,..v~) = u’;(x,..~.), The fullrange double Fourier
(Al)
series expansion for the function and its two first partial derivatives are as follows
wherein
h
u’~(.~,..~:)cos(r,.~,)sin(~..~:)ds,
ld’:(.r,..v:)
sin (/I..v:) ds, d.v:
d.r,
for
for
n = I. 2,.
m.n=
. . . IX)
I.2 ,...,
a~
(A3a)
(A3h)
:
R. A. CWAL.DHURI and K.
R. ABCARIA
(A3e)
(A3d)
(A3ei
(AX1
1A3g)
Integration
by parts 0f’r.h.s.
of eqn (A3a)
will yield the following:
x sin (/j”.v:) d.r,
for
m.n=
c ,I
,.h~O):(l)"cos(l,.s,)ds,+~,~
1.2 ,...
.r:
(A4n)
,I’ ’ 3
2 (u:‘(.r,..r,0)
‘G If=
I
that if;, function ~(x,,s~) is an odd function of x,. i = I, 2, reprcscnlcd by i( fullrange double Fourier series in the dom:lin ( tr.~) x (/I./J). krmwisc parGal ditkrcnliatlion of the Fourier series with respcvt to .r, does not rcprcscnt the lirst partial dcrivxtivc F,(.Y,,.v~) unless (i) F,,(.x,.s~) is conGnuous in the interior of the ubovc domain. and (ii) F(r)0.x2)
= F(tr+O..r,)
= 0
In the case of an cvcn function. condition dcrivntivcs. as is cvidcnt from cqns (A4).
for
i =
I and/or
F(s,.h0)
= F(s,.
h+O)
(i) will suHice. This is clearly true for u:‘( r ,.x2)
for
i = 2.
(AS
and its tirst pnrtial
The prcscnt solution. c.g. u’~(.x,,s~) [see cqns (IO)), is represcnkd by douhlc Fourier scrics in the domain (0.~1 x (0, h). the lengths of the mtcrvals in the directions. .r, nnd s2, b&g onehalf of the full ranges of inkrvals ofpcriodicity, Ztr and YJ. rcsptakvcly. u~(.v,, x?). in addrlion to being an cvcn function of x, and an odd function of x2 3s staled carlicr. is also continuous in the interior of the domain (0.~) x (O,h), and dots not vanish at the cdgcs. s: = 0. h. The halfrange double Fourier scrics repre~n~~Ition for &(x,.x2) and its tirst two partial derkativcs arc given by eqns (AI ). whcrcin
u’~,:(\,..r:)cos(z,.~,)cos(P,.r:)d.r,d.r:
~r:‘~(_~,..\.:)cos(P~.~.?)d.~,
ds:
for
for
R = I. 2..
nr.,t = I.:! ..,..
.x
z
(MC)
Stnt~c analysis of cylindrical
Substitution
I5
panels and shells
of L/“”
into eqn (AL)
= I:
xld = 0
and
u’/(O0..~~)
= u’,‘(O+O. x:)
(A7)
will yield
B_ = x,L’;,.
(AR)
which implies that the tirst partial derwative. u’,‘,, (.t,. r ?). can be obtained by term%ise differentiation of the halfrange Fourier series expansion of u’/t.~ ,..r:). represented by eqns (Ala). (Ahu). However. the other first partial derivative. u’,‘,.(.Y,. .x2). cannot be represented by the termwise diffcrcntiation of the series. given by eqns (Ala) and (A6a). because u’,‘.:(.Y(..v~) has ordinary discontinuity at the lines.’ = 0 and also because u:‘(x,.h0) The Four&
coefkients
= , .
# u:l(.v,. h+O)
for 11’;:( Y ,. 1:) must then be obtained _k:,, = 0
and
u’,‘(.v,. h+O)
# 0.
(A9)
by substituting in eqns (Al)
= u’:(.v,.h0);
u’~(r,.O0)
= rc’~(.~,.O+O)
(AIO)
which will finally yield
Cl”
= ,,,(.;,
+ !_ I’ f,~,,(.~,.~~)(I)“u’~(~,.O+()):COS(~,.I,)~.\., ” trh s0
(“’n,o = P. ( ‘L!”+ 5
(Ir:‘(r,.hO)rc:‘(r,.O
to); cl)s(l,r,)dr,
I”I,
AI’I’FNDIY .
I
fl
Ik/;/riliolr Ol/‘l’Cr,t,,,, l’fl,r.rlcotlr The nontcro consktnts rsfcrrcd to in cqn (6) arc written as f~~llows:
for
for
01. it = I. 2.
,,I = I .?.....,r
, r,
(Al la)
(Al Ih)