Static analysis of moderately-thick finite antisymmetric angle-ply cylindrical panels and shells

Static analysis of moderately-thick finite antisymmetric angle-ply cylindrical panels and shells

STATIC ANALYSIS OF MODERATELY-THICK FINITE ANTISYMMETRIC ANGLE-PLY CYLINDRICAL PANELS AND SHELLS REAZ A. CHAUDHURI Department and KAMAL R. ABU-ARJA+...

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STATIC ANALYSIS OF MODERATELY-THICK FINITE ANTISYMMETRIC ANGLE-PLY CYLINDRICAL PANELS AND SHELLS REAZ A. CHAUDHURI Department

and KAMAL R.

ABU-ARJA+

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)

Abstract-The analytical (exact in the limit) or strong (or differential) form of solutions to the benchmark problems of(i) axisymmetric angle-ply circular cylindrical panels of rectangular planform and (ii) circumferentially complete circular cylindrical shells. subjected to transverse load and with SS2type simply-supported 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 boundary-discontinuous double Fourier series approach. for three kinematic relations. which are extensions of those due IO Sanders. Love and Donnell to the first-order shear deformation theory (FSDT). Numerical results presented for twolayer square antisymmetric angle-ply 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 bending-stretching

1987). Confidence

in these approxi-

mate numerical techniques is dependent upon the closeness of agreement of the numerically predicted displacements and stresses of certain bench-mark counterparts. cylindrical

Thin

or moderately

problems with their analytical

thick cross-ply and antisymmetric

shells and panels, with certain types of simply-supported

angle-ply

circular

boundary conditions,

are two excellent examples of such bench-mark problems, which have attracted considerable attention in recent years. Closed-form

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 Abu-Arja

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 non-axisymmetric

mation, buckling and vibration of cross-ply circular (circumferentially) shells and panels, with SS3-type [under the classification

(1975),

Soldatos and defor-

complete cylindrical

of Hoff and Rehfield (1965) and

Chaudhuri ef al. (1986)] simply-supported boundary conditions prescribed at the edges. Soldatos (1984) has used a Flugge-type theory and Galerkin’s approach in solving the free vibration

problem of non-circular

cross-ply cylindrical

t Present address: LCC Siporex. P.O. Box 6230. Riyadh

shells.

11442. Saudi Arabia.

R. A. CHAUVHLKI and K. R. Am-ARJA

2

With regard to the problems of antisymmetric (1983a.b)

and Whitney

angle-ply cylindrical

(1984) have resorted to Galerkin’s

panels. Soldatos

method. Soldatos (1983b) attri-

butes the non-existence of an exact solution for an angle-ply 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 closed-form

While this statement does not attribute impossibility

solution seems to be impossible”. obtaining

angle-ply 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 cross-ply laminated

shells. and has concluded

that “Closed-form

solutions for deflections and natural frequencies of simply supported, cross-ply laminated . . . shells are derived . . . . Unlike plates. antisymmetric angle-ply laminated shells with simply-supported boundary conditions do not admit exact solutions. The exact solutions presented herein for cylindrical and spherical cross-ply 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 cross-ply 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 cross-ply 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 ( 107-l. 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. Closed-form

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 boundary-discontinuous

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 doubly-curved

difTcrential either in the

depending on the coctlicients of the

(iii) ensures uniqueness of the solution,

leads to a highly efficient computational

moderately-thick

partial

of antisymmetric

anglc-ply

cylindrical

Ana-

panels and cir-

complete cylindrical shells have, however. not been investigated in detail and

their numerical results are still non-existent. The primary objective of the present study is to apply the aforementioned in obtaining a unique solution to the five highly coupled second-order

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 bench-mark 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 first-order

shear deformation

results for antisymmetric

theory (FSDT)

and (ii) some useful numerical

angle-ply cylindrical panels limited to Sanders’ kinematic relations

alone. as a first step. Although

the problem of doubly-curved

anglc-ply panels was solved

Static analysis of cylindrical

panels and shells

3

by Chaudhuri and Abu-Arja (1988) using a preliminary version (Chaudhuri. 1987) of the boundary-discontinuous 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 angle-ply 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 bench-mark 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 (Raleigh-Ritz. 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 angle-ply 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 mid-surface 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. ABU-AIUA

N,,, +Nc-- C’2M6.2/R= 0: NdR--Qt.,

-Qz.z-q

= 0;

N,, + FZM6., lR + N2,2 + C,QJR = 0 ;

M,,,+M,.2-Q,

= 0;

M,.,+IL~:.~--Q~

=0

(3)

in which q is the transverse or radial distributed load. Surface-parallel stress resultants, N,. stress couples (moment resultants), Mi, and transverse shear stress resultants, Q,. are related to the mid-surface 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 angle-ply 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 SS2-type 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 well-posedncss 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 SSZ-type simply-supported 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. Am-ARJA

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 functions-uj 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) m-l “-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\:‘)

m-l

-

-

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. Aeu-AIUA

!!

(Usl’ -/g-u:?‘) 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. angle-ply 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

plate-eqns

(8).

(A I) of Chaudhuri and Kabir (1989). who have presented solutions only for the SSJ-type simply-supported amI C4-type clamped boundary conditions. However, solution in the form of linear algchraic equations for the SS4-type boundary conditions can easily be modified to obtain their SS2-type 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, moderately-thick

cylindrical

RESULTS :ISan

AND

DISCUSSIONS

example, a two-layer antisymmetric

panel of square planform

angle-ply

(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 fiber-reinforced

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

non-dimensionalized

(Chaudhuri

identical material and Abu-Arja,

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 mid-points of the two sides, (u/2.0) and (O./J/~). respectively. It may be noted that not only the side-to-thickness 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. ABC-ARJA

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.x-Y6

I.YII 13.75

IO

$4

4

8

6

4

m = n

u:

22.5

0

of two moderately-thick

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 SE-type

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 surface-parallel

tiber orientation

cylindricul

the variation with the tibcr orientation

pitrallcl displaccmcnt, II:. and the rotation, &z, at the mid-point axis. (trj2.0);

results

displacements. rotations

_r: = h/2. of ;I modcratcly-thick

(U//I = IO) with Cl= 45 ‘. Figure 4 exhibits of the non-dimcnsionalizcd

( 19X9). Numcricnl

have been obtained by using 111= N = 8.

f:igures 2 :IIICI 3 show the variation ofthc non-dimcnsionnlizcli 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 surface-parallel displacements and rotations are antisymmetric with respect to the same. Similarity of these plots to their independently computed spherical [see Figs 2-S. Chaudhuri and Abu-Arja (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. AK-ANA

6. COSCLUSIONS

The analytical the problems

(exact

in the limit)

of moderately-thick

circumferentially

complete

developed

boundary-discontinuous

novel

facilitates

circular

the well-posedness

coefficients

of the assumed

boundary-discontinuous equal to the number mark problems be incapable

double

of unknowns

techniques

as the finite element

to the first-order 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 side-to-thickness response of these pan&.

solutions

(FSDT).

angle-ply

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

two-layer

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) angle-ply

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 Ahu-Arj;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 ;~nglc-ply 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 modcrarcly-thick 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 boundary-discontinuous double Fourier srrics solution to a nybtcm of complelcly coupled PDEs. /III. J. Engng Sci. 27, 1005-1022. Chaudhuri. R. A. and Abu-Arja. K. R. (IYXX). Exuc~ solution of shear-tlcxiblc doubly-curved ;tn~isymmrtric ;tnyle-ply shells. brr. 1. Engng .Sci. 26. 587-604. 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 boundary-value problems of douhlycurved moderately-thick orthotropic shells. Inr. J. Engng Sci. 27. 1325-l 336. Chia, C. Y. (IYXO). Nonlincwr Analysis of Plures. McGraw-Hill. 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. IOYI-1097. 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. (194-I). 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. 15--B. llirano. Y. (lY7Y). Buckling of angle-ply 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. 541-546. 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. 155-l 77.

Static analysis of cylindrical

panels and shells

13

Hughes. T. J R. ( 1987). The Fimrr Ekmtwr .Wuthod. Prentice-Hail. Englewood Cliffs. New Jersey. Jones. R. XI. and !Uorgan. H. S. ( 19751. Buckling and vibration of cross-ply laminated circular cylindrical shells. .41.4.4 II 13, 664 671. Kdhlr. H. R H. .md Chaudhurl. R. A. ( I9911. Free vibration of shear-flexible antisymmetric angle-ply doubly cur\ed pan&.

Inr. 1. Sditb

Smccrurc~

28, l7--32.

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. 6.94-l 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. 1563-l 579 SlnhJ. P. K. llnd Rath. A. K. (1976). Transverse bending of cross-ply laminated circular cylindrical plates. J. I/~YA. Enyrtu 5c.i. 18. 53 56. Soldatos. K. P. (lY83a). On the buckhng and vibration of antisymmetric angle-ply laminated circular cylindrical shells. Inr. J. Engn,g Sci. 21. 217-212. Soldatos. K. P. ( 1993b). Free vibrations of antisymmetric angle-ply laminated circular cylindrical panels. Q. J. .Mtz,h. Appl. .Vurh. 36. 207-?,I. Soldatos. K. P. (IYYJ). A Flugge-type theory for the analysis of anisotropic laminated non-circular cylindrical shells. Inr. 1. Solitls SIIIICIW~.~ 20. lO7-120. Soldatos. K. P. ( 1987). Influence of thickness shear deformation on free vibrations of rectangular plates, cylindrical panels and cylinders of antisymmetric angle-ply construction. /. Sound V&r. 119. I I l-137. Soldatos. K. P. and Tzivanidis. G. J. (1982). Buckling and vibration of cross-ply 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. Prentice-Hall. 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.

Mcc-h. 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 full-range 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. ABC-ARIA

(A3e)

(A3d)

(A3ei

(AX-1

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( full-range 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,.h-0)

= 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 one-half 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 half-range 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 (A-L)

= I:

xld = 0

and

u’/(O-0..~~)

= 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,.h-0) 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 (A-l)

= -u’:(.v,.h-0);

u’~(r,.O-0)

= -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,.h-O)-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 non-tcro 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)