Three-dimensional elasticity solution for static response of orthotropic doubly curved shallow shells on rectangular planform

Composite Structures 24 (1993) 67-77

Three-dimensional elasticity solution for static response of orthotropic doubly curved shallow shells on rectangular planform Alavandi Bhimaraddi* Dwerst~ed Computer Engmeermg and Development, Clawson, Mgch~gan,USA The analysis of homogeneous and laminated doubly curved shells made of an orthotropic matenal using the three-dunensional elasuc~ty equatmns is presented. Solution is obtained by uuhsmg the assumpUon that the ratio of the shell thickness to its rmddle surface radius is neghglble as compared to umty. However, it is shown that by &vading the shell ttuckness into layers of smaller thickness and matching the interface displacement and stress contmmty con&tlons, very accurate results can be obtained The two-dlmens]onal shell theones are compared for their accuracy in the hght of the present three-dmaensmnal elasucity analysis. Numencal results for orthotrop~c shells show that the twodunenslonal shell theones are very inacurrate for shells with thickness to length ratio greater than or equal to 1/10.

INTRODUCTION Most practical problems, in general, are threedimensional (3-D) in nature. Because of the extremely complex nature of the 3-D elasticity equations rigorous solutions are available for only a few problems. However for many practically important problems usable solutions have been obtained. For example, the 3-D solution for isotropic plates has been presented by Srinivas e t al. 1 and for orthotropic plates by Srinivas and Rao. 2 It is clear from these works that the resulting equations are differential equations with constant coefficients whose solution is simple and straightforward. The presence of curvature as in the case of doubly curved shells poses additional difficulties when the 3-D equations are employed since the resulting equations are differential equations with variable coefficients. The exact solutions to such equations have been given by Greenspon 3 and G a z i s 4 for the case of free vibration of an isotropic closed cylinder; and by Chou and Achenbach 5 a n d S r i n i v a s 6 for the case of free vibration of orthotropic cylinders. The static problem of an isotropic long cylinder subjected to general kmds

*Present address: 27000 Franklm Road, Apt 206, Southfield, M148034, USA.

of surface loads has been considered by Fliigge and Kelkar 7 and Yao. 8 To avoid the inherent mathematical difficulties the 3-D problem is reduced to a one-dimensional or to a two-dimensional problem depending upon the geometrical dimensions of the structure. For example, shell problems are reduced to 2-D problems due to their smaller thickness in relation to their other linear dimensions and utilising certain hypotheses regarding the kinematics of deformation. In the classical shell theories (CST) the thickness shear deformations are considered to be negligible as a consequence of the assumption that the normals to the middle surface remain straight and normal to the middle surface before and after deformation as may be found in the monumental works of Fhigge, 9 Gol'denveizer, ~° Sanders, ~ and Timoshenko and Woinowsky-Krieger. 12 The shear deformation theories based on the hypothesis that the normals to middle surface before deformation remain straight but no longer remain normal to the middle surface after deformation. The theories based on this hypothesis are attributed to Mindlin, 13 Mirsky and Herrmann, 14 and Reissner. 15 There are a number of higherorder theories which disregard the KirchhoffLove hypothesis completely and assume that the normals do not remain straight nor do they remain normal to the middle surface after deformation. Discussion of such various theories can

67 Compostte Structures0263-8223/93/S06.00 © 1993 Elsexaer Sc]ence Pubhshers Ltd, England. Pnnted m Great Bntam

68

Alavandt Bhtmaraddt

be found m the works of Bhunaraddi ~6 and Stein. ~7 In short, it may be said here that most of these 2-D theories fail to satisfy the interface transverse stress continuity conditions m the case of laminated structures. Since the interlammar stresses have an important influence on the onset of delamination and the consequent failure of the laminate, the knowledge of these stresses is essential and important. The correct estimation of these transverse stresses can only be obtained by employmg the 3-D equations in the analysis. Hence, m spite of the analytical difficulties involved in the analysis of doubly curved shells using the 3-D equations, any effort m solving these problems Is of considerable Importance. In addmon, the results from such an analysis are useful in validating the less approximate 2-D shell theories Recently Bhimaraddi and Chandrashekhara ~8 have presented a 3-D solution to simply supported orthotroplc, homogeneous and laminated cylindrical shells. In that study the differential equations w~th variable coefficients were first reduced to those w~th constant coefficients by utdlslng the assumption that the thickness-toradms ratio is negligibly small and then by diwdmg the thickness into a number of smaller thickness layers very accurate results were obtained. A similar analytical approach has been extended in this paper to the analysis of doubly curved shells on rectangular planform.

BASIC EQUATIONS OF THE THREEDIMENSIONAL ELASTICITY For an open shallow shell the maddle surface can be defined by a set of cartesian coordinates as it differs little from a plane surface. Assuming the twist of the surface to be zero, the strain displacement relations of the 3-D elasticity equations corresponding to the present problem (referrmg to the cartesian coordinate system of Fig. 1) are written after modifying the general relations of Saada 19 as

R1 l ou+wl. 0v+ w ] .

x[_~y R-~2J'

(R21

0w[R]

× -- +

--,

0y

0x

R 1

y~z =

O z ?YZ=

x

(I) R2

0Z

In the above equauons x, y, z are the cartesian coordinates m which z is measured from the middle surface of the shell; u, v, w are the displacements in the x, y, z-directions; R 1 and R 2 are the middle surface radii of curvature (which are assumed to be constant); ex, ey, ez are the normal strains in the x, y, z-directions; ~xy, ~lxz, ~yz are the shear strains. In order to reduce the system of equations with variable coefficients to those with constant coefficients the only assumptions we need to make are the following:

In order that these assumptions are indeed true, it is essential that [h/R~] and [h/R2]< < 1. Equataons (2) can easdy be satisfied if the shell is slightly curved or if the thickness of the shell is very small compared to the radii of curvatures. Thus in the case of thick shells, the thickness of the shell is to be divided into a number of layers with smaller thickness so that eqns (2) are satisfied. This will allow us to obtain the exact values

~

x (uJ

z O)A

~

~

/

.

a

~ ~ 7 Thickness Defods for Laminated Shell

y

e

r

1 Loyer 2

~ - Shell middle ~'~ surface Layer n

Fig. 1. Dmaenslons and coordinate system for doubly curved shallowshell.

69

Analysis of orthotropic doubly curved shallow shells

for thick shells. Utilising eqns (2) in eqns (1) we write the strain-displacement relations as Ou

av

w

ex----i+-- ; Bx RI

ey =

three displacements as O2U

w

+--; Oy RE

on[2 1]c44

X----

Ow F'z --- B---z

BZ

Ov Ou YxY = -~X 4 Oy '

Ow Ou

Ov

u

R1

u+[C66..I-C12]~

R2 R1

(3)

v

logI roll c12c131[x] o~ =/Cl~ C~ C2~| e~ Oz LC13 C23 C33] Ez

OxOz

~'xz = C44~xz;

C66 Bx-------~ -~-C22 ~ -~-C55 ~ ~t-l - - -iByE Oz [ R1 RE J

Bv [ 1

2]C55 B2u V'[-[C66"{-C12] R1 R2 -~2 BxOy

X~-- - - + - Bz

(4a) + "C22 + 2 C55 R2

c,: +___C,,]Ow R1

Here, ox, By, oz are the normal stresses; Zxy, rxz, Zyz are the shear stresses; C,j are the elastic constams of the orthotropic material. Using eqn (2), 3-D stress equilibrium equations (Saada 19)can be written as

~2 W

(6)

x~=O

OyOz

.El3 - Ell -- C44 C13- C12] OU

B2u 2 +_;.__] 1 r~z= 0 BOx+ B_F_~+ Or= + [_2_

Dx

By

Oz

LR1 K2J

OZ

+-R1 R2

Br=+_z_mBr+Bez+ Bx 0y Bz

ox o~= 0 R1 RE

j -~y +[CEa + c551

~'yz : C55Yyz

(4b)

By

BxBy

ev

The stress-strain relations for an orthotropic material read

Ox

02v

2 Ow x~=0

YYz-----~y+~Zz" R2

"gxy: C66Yxy;

--'l"--

Cll + 2644 Cl 2 "t- C44] Ow + RE J-~x +[C13+ C44] + R1

Yxz=~x+~z-R-- ; Ow

B2U I [ C 4 "C441 4+

B2U

CI1-~-~x2+ C66v+ C44-O~z2+L--~I --R-~2 j

B~----z

oxoz

+

+

B2w

x ==-~ + [ c . + c . ]

r~=0

+[C13

C23-C22- C55 f C23.~-F12] BV RE R1 J By

02w 02w 02w + 644 0x 2 ~-C55 ~ByE+ C33 Bz2

[1 + 1] L~ ~J o, (5)

Subsituting the stress-strain relations (4), via strain-displacement relations (3), the above stress equilibrium equations can be written in terms of

+LR, R2] C33 Oz L R1 + C13 + C23 - 2C12 R1R2

+

C23-c22] R2 ] w=0

C44]

A lavandt Bhlmaraddt

70

The above equations are the required equilibrmm equauons and they are the differential equations with constant coefficients. Had we not made use of the assumptions in eqn (2) the above equilibrium equations would have had coefficients involving the normal coordinate z, the solution of which would have caused a great deal of mathematical difficulty. In the next section the solution of eqns (6) is sought using the method of separation of variables.

SOLUTION

OF

THE

EQUILIBRIUM

EQUATIONS

Solution of the equilibrium equations (6) is difficult to obtain for given general boundary and surface condltions. However, all-round simply supported shells render possible the solution of these equations m terms of trigonometric series. The following series solution for displacements and stresses satisfy the simply supported boundary conditions:

sin Ny sin Ny

u = X X Umn cos Mx sin Ny;

O r = • • Sxm n sin M x

v= X X Vm, sin Mx cos Ny;

(YV-~"~" ~" S~,mn sin M x

w = Z X Win, sin mx sin Ny;

O. = X Y~ Szmn sin Mx sin Ny

r~== X X T~,zm,,cos Mx sm Ny;

rv, = ~, X Tyrmn sin Mx cos

(7)

Ny

"t;~,= ~. E TrvmnCOS Mx cos Ny

where M = maria and N = n ~ / b Here a and b are the dimensions of the shell in the x- and y-d~rections; m, n are the summation mtegers. In the above series solution we note that Urn,,, Vmn, and Wm,, are functions of the normal coordinate z and are to be determined as the solution of the following ODEs, which are obtained after substituting eqns (7) in eqns (6) as LllUmn + L12Vmn + L13 Wmn=O

(8a)

L21 Urn.+L22 Vmn+ L23 Wren= 0

(8b) (8c)

L31Umn + L32 Vmn + L33 Wmn= O

where the differential operators L,j are given by d2

d

L~I = a~ - -

--

dz 2 + a 2 dz +

a3 ,

L12

•

=

d L13=a5 d---~a6;

L21 = a 4 ,

.

d

d2

L22 = a7 dz2+ a8

z+a9 (9)

d L23 = alO -~Z+ a11,•

L3I

d a12 ~z+ al~ ",

=

d L32= a14 ~zz+ als ,"

d2 - + L33 = a16 -dz2

d -a18 a17dz+

al-al8 appearing in eqns (9) have the following definitions: =

a1=C44,

a4=

a2

[1+11 JR1

-- [C12 + C66] MN;

a 7 = C55;

RzJ

C44;

a3=--[

a5 = [C23 + C44] M;

a 9 = --

2 + 1 C44 --] - C l l M 2 - C 6 6 N2 ~ R2 RI

Cll "~-2C44 a6=

+

R1

C12 --I-C44]

+ nL jM

c55-c66 2-c22N2 R2

Analysis o[ orthotrop& doubly curved shallow shells

[C22+ 2C55 C12 + C55.] all= L R2 + gl / N;

alo = [C23 "~ C55]N;

a13

a15

Cll + 644- C13 ~-C12 - C13 ] M; = gl /~2 J

=[622 "~-655 - C23 ]- c1

7c

RE

R1

a18

[ Cl . [

R 1

] N; J

)

R1R2

d

a12 --

71

-- [C13 "1"C44] M

(10)

a14--- [C23 -t- C55] N

a16-- C33 ;

[1 1]

a17= _ _ + - -

R1

R2

C33

_ C . . M 2 - C55N 2

R2

_]

The solutions of eqns (8) are obtained by expressing Umn, Vmn, and W,.n in terms of displacement potential function ~,.n as follows

U,. =[LI2L23-L,3L22]¢..;

Vmn--[L13L21-L23Lll]¢mn; Wmn=[LI1L22-L12L21]¢mn

(11)

Substituting the above solution in eqns (8), it can be seen that eqns (8a) and (8b) are satisfied identically and eqn (8c) reduces to the following governing equation in #m.:

d6 ¢ .~t. d5 ¢ .~t_c3 d ' ~ , c4 d3¢+ c5 dZ~+ c6 d~+ c7¢ ~--0

Cl dz 6

c2 dz 5

dz 4

dz 3

dz 2

(12)

dz

In the above equation and in the subsequent analysis subscripts 'mn' have been dropped for the sake of simplicity and the expressions for cl to c7 are given in the Appendix. Since eqn (12) is of order six, the solution ¢ involves six arbitrary constants and is sought in the form ¢ = e az

(13)

and six a values are obtained as the roots of the following equation Cl a 6 + c2 a5 + c3 a4 + c , a 3 + Csa 2 + c6a + c7 = 0

(14)

The exact expression for qt depends on the nature of the roots of eqn (14), for example, for six real and distinct roots we can write ¢t= A ~ea'z + A2ea2Z+ A3e a3z+ A4e a,z + A5e~SZ+

(15)

A6ea6Z

Here, A l - A 6 are six arbitrary constants to be determined from the given surface traction conditions at z = + h/2 as ryz= qyl;

rxz = qx~;

trz = qzl (at Z= + h/2);

rxz = qx2;

ryz = qy2;

trz = qz2 (at Z------ h/2) (16)

In the above qxl, qy~, qzl are the given applied tractions on the outer surface of the shell in x-, y-, zdirections and qx2, qy2, qz2 are those on the inner surface. Once ~ is obtained from eqn (15) the displacements U, V, and W can be computed using eqns (11); strains can be computed from eqn (3); and the stresses from eqn (4) as

F¢I'7 U

Txy

=

b5

b6

b7

b9bxobll

b8

b12 ~

I

(17a)

~

72

Alavandt Bhtmaraddt

TT,; lb,, Lb2

(17b)

b21, b,, b2:l o,ll ¢i I

b~4 b~5 b36 b37 b38 b~9

]411 1 I._~

(17c)

._1

In eqn (17) the superscripts indicate differentmtion with respect to z and the expressions for b 1-b45 are given in the Appendix. This completes the formulaUon of the problem and Rs solution for a shallow shell on rectangular planform with given geometric and material parameters [R 1, R2, a, b, h and C,j]. It should be pointed out here that the above procedure holds for one layer in the case of laminated and sandwich shells. Thus, for each layer there are six arbitrary constants to be determmed to obtain the complete soluuon for that layer. These arbitrary constants can be obtamed using the interface stress and displacement continuity conditions in addition to the known surface traction conditions. For layers i and l + 1 the interface conditions are

(18)

For example, if the shell is made up of two layers there will be 12 constants (A j-A 12, six for each layer) to be determined using the six surface tracUon conditions (see eqns (16)) and six interface (three transverse stresses and three displacements) contmmty condiUons (see eqns (18)). Before presenting the numerical results we note that when the ratios h/R~ and h/R2 are not small enough to be considered as < < 1, the thickness of the shell is divided into a number of layers with smaller thickness values so that for any layer the value of the ratio h/R~ and h/R 2 becomes < < 1 and the solution is obtained using the interface stress and displacement continuity conditions (18) and the surface traction conditions (16). For laminated shells and sandwich-type shells for which h/R~ and h / R 2 are not too small, each layer shall be dwided into sublayers with sufficiently smaller h/R~ and h/R2 ratios. However, a proper value for h/R 1and h/R2 can be chosen by conducting some numerical experiments which are demonstrated in the next section.

Further, it is to be noted here that the corresponding equations for rectangular plates can be deduced from the present analysis by using h/R 1 = h/R2=O and those for cylindrical shells by using 1/R~=0 and R2=R. Thus, the present procedure and the corresponding computer coding has been checked by computing the results

for lsotropic (Srmivas et al)), orthotropic plates (Srimvas & Rao 2) and cylindrical shells (Bhimaraddi & Chandrashekhara~8). Also we note that the approximations made in eqns (2) hold true for rectangular plates and hence exact results can be obtained without dividing the thickness of the plates.

Analysis of orthotropic doubly curvedshallowshells

Table 1. Convergence studies on an orthotropic spherical shell with different h/a ratios (_RI=Rz=R, R/a=l,

DISCUSSION OF NUMERICAL RESULTS Numerical results can be obtained for any applied load on the shell surfaces by expressing it in the form of a double Fourier series and considering as many number of terms as required from the viewpoint of convergence. However in this study we restrict ourselves to a simple load case which allows us to consider one term in the double Fourier series of eqns (7) with m=n = 1. Thus, numerical results are obtained for sinusoidally distributed lateral loading on the outer surface [qxl ~" qx2 -~- qyl = qy2 = qz2 = 0] o f t h e shell as

q~l= Q sin zrx / a sin ~y / b

a/b= l) (fC=WE,/Q,.T~=T~/Q; -x

.

.

-t.

S,,= Sx/Q) .

Nd"

h/a=O'l

1 2 5 10 15 20

4"2715 34-989 24899 4-5876 35"094 23423 4-6958 35"678 2"4436 4"7117 35 318 2.4371 47147 35 172 2"4415 41757 35"109 24402

.

.

(Ex= 25Ey; .

0"4)

h/a=O.15

2.2711 24730 25522 25641 2-5664 25672

21064 22 285 22181 21"938 21-832 21775

2"0674 1-9383 20731 20671 2"0730 2"0715

"Number of dwlsions in the shell thickness

Table 2. Convergence studies on an orthotropic spherical shell with different R/a ratios (RI=Rz=R, h/a=O.l,

a/b-- 1) Nd

Ex=25;

E~=I;

G~z_G~

Ev

Ev

Ey

Ey

R/a = 5

1 2'

G~

/txY= 0"25;

-x

(19)

The Poisson's ratio has been assumed to be 0.3 in the case of an isotropic material and the orthotropic material properties correspond to

1 Ey = 5 ;

73

/Z~x=0"03;

/~y~=0.4 Here Ex, Ey, Ez are the Young's moduli; Gxy, Gxz, Gyz are the shear moduli; and Pxy, Pu,/~yz are the Poisson's ratios. Table 1 shows the convergence studies on a homogeneous spherical shell with two h/a ratios. It may be seen from this table that as the h/a ratio of the shell increases one has to divide the thickness of the shell into a greater number of layers to achieve the convergence of the results. As may be observed from Table 2, this is also the case for decreasing R/a ratio. For a shell with h/R=O.O1 (h/a--O.1 and R /a= 10) the exact results are obtained with N o -- 1 as may be inferred from Table 2. Thus one can conclude from these tables that if h/R ratio of the shell is ~<0.01 no division of the thickness of the shell is required. It may also be noted that for layered shells with h/R
R/a = 10

Lz 1 2 5 10 15 20

6"2881 6"3216 6"3318 6"3332 63335 6"3336

56 398 56"605 56620 56"605 56597 56"593

4"1720 41758 4"1850 4 1852 4"1855 41855

L 6"3477 6"3563 6"3589 63593 6"3593 6-3593

56"949 57"002 57"006 57"002 57"001 57"001

4"2262 4"2273 4"2296 42297 4"2297 42297

orthotropic shell considered here the deflections are far less than those for an isotropic shell of identical geometric parameters. For the sake of comparison of the 2-D shell theories with the present 3-D analysis, the shallow shell versions of the parabolic shear deformation (PSD) theory as proposed by Bhimaraddi 16,2° and the classical shell (CST) theory, which is the special case of PSD, have been used. The difference between the 2-D shell theories and the present 3-D solution is quite considerable in the case of thick shells. It may be noted that the 2-D shell theories predict lower deflection values when compared with the 3-D analysis. For example, for an isotropic shell with R/a = 1, the error in PSD is about 1% for h/a=O.01 and 14% for h/a=0.1 and similar errors for CST are 1% and 15%. It may be observed from this table that the errors in 2-D theories are higher for orthotropic shells as compared to isotropic shells. The PSD gives much better results than the CST when compared with the 3-D analysis, especially so in the case of an orthotropic shell. However, marginally better results could have been obtained from the 2-D shell theories had we employed deep shell theory

74

Alavandi Bhimaraddt

Table 3. Comparison of centre deflection ($~) for homogeneous spherical shell with different h/a and R/a ratios (a/b= 1, RI =R2--R)

R/a

Isotroplc shell*

h/a=O.O1

Orthotropic shell

h/a=O 1

h/a=0.15

87095 74751 73702

4 9497 3-8929 3"6979

h/a=O01 75397 74207 74200

h/a=O 1

h/a=O 15

47117 3-4553 27447

25641 17229 10191

3-D PSD CST

100.59 99.645 99.644

3-D PSD CST

39645 394 37 39437

18451 17"013 16"480

7.7240 6 9261 6 3322

28572 282.32 28224

59693 5 2610 37736

26788 2 3180 12015

3-D PSD CST

87536 872-02 87200

23"381 22277 21371

85912 8"0940 72945

59343 587 39 58700

6.2215 5 8248 4.0551

26635 24764 12427

953.25 944.66 94366

6 3014 6 0517 41638

2.6494 2.5371 12578

3-D PSD CST

15183 15136 15136

25 785 24 983 23"849

89235 86017 77043

3-D PSD CST

23014 2295-4 2295 3

27061 26 471 25"201

90755 8-8589 79099

1325 5 13148 1312.9

63332 6 1629 4 2161

26393 25662 12649

10

3-D PSD CST

73831 73713 73702

28910 28754 27 262

92505 92267 82019

27677 2753.2 27447

6 3593 6 1376 4 2880

26256 26061 12745

20

3-D PSD CST

16499 16485 16479

29"356 29"388 27 831

9"2666 93235 8.2783

38025 37897 37736

6.3532 6 3575 43063

2.6022 26162 12770

O0

3-D PSD CST

29504 28041 28026

29440 29-606 28026

92352 9.3562 83040

43430 43335 43125

6 3343 6 3709 43125

25879 26196 12778

3-D -- Present 3-D Analysis, PSD -- Parabohc Shear Deformation Theory, CST -- Classical Thin Shell Theory, * Polsson's Ratio = 0 3

equations instead of the present shallow shell versions. Table 4 shows centre deflectaon values for laminated spherical shells with two different lamination schemes. It is evident f r o m this table that the deflection values for the 0/90 a r r a n g e m e n t are less than those of 0 / 9 0 / 0 a r r a n g e m e n t and a h o m o g e n e o u s shell. This observation ~s a direct c o n s e q u e n c e of the presence of b e n d i n g - s t r e t c h ing coupling in the 0 / 9 0 a r r a n g e m e n t m addition to the usual b e n d i n g - s t r e t c h i n g coupling arising out of the presence of curvatures in the g e o m e t r y of the shell. O n c e again we observe for laminated shells that the 3-D analysis predicts higher deflection values than the 2-D shell theories. T h e errors in the 2-D theories are slightly less in the case of 0 / 9 0 shells are c o m p a r e d to the c o r r e s p o n d i n g 0 / 9 0 / 0 or h o m o g e n e o u s shells.

CONCLUSIONS T h e three-dimensional elasticity solution for doubly curved shells m a d e of an orthotropic material has b e e n presented. Using the assumption that the thickness-to-radius ratio is negligible c o m p a r e d to unity, the governing equilibrium equations have b e e n r e d u c e d to differential equations with constant coefficients which would have otherwise b e e n those with variable coefficients. Further, by dividing the shell thickness into sublayers such that their individual thickness-toradius ratio is kept as low as practicable (and in this study it is shown to be 1/100) very accurate results are obtained for shells whose thickness to radius ratio is not negligible c o m p a r e d to unity. N u m e r i c a l results for orthotropic shells indicate that the differences in the predictions of the two-

Analysis of orthotropic doubly curved shallow shells Table 4. Comparison of centre deflection (I~) for cross-ply spherical shell with different Ri = Rz ---R equal thickness plies)

R/a

0/90 arrangement

3-D PSD CST

4

75 h/a

and

R/a

ratios

(a/b= 1,

0/90/0 arrangement

h/a = 0.01

h/a = 0 1

h/a= 0.15

54"129 53"503 53"493

4"6920 3"7686 3"5718

2"7386 1"9581 1"6769

h/a-- 0.01 54.252 53.491 53.486

h/a = 0.1

h/a-- 0"15

4.0811 3.0770 2.4008

2.4345 1.6564 0.9438

3-D PSD CST

212"33 210"84 210-79

8"8092 7"8119 7"1163

3"8190 32487 25835

20836 206-34 20627

6.3134 5.3616 35965

3-0931 25253 1.1739

3-D PSD CST

456"46 462"93 462"82

10"512 9-7489 87192

4"0856 3"7004 2"8709

441"81 438.23 437.92

6"9888 6.2163 3.9619

3.2228 27970 1.2294

3-D PSD CST

799.81 796.07 795'86

11"263 10"675 9"4655

4 1758 38897 2-9872

727-62 72237 72154

7-7476 6"5836 41080

3.2605 2-9065 12501

3-D PSD CST

1198.7 1193.7 1193.3

11"639 11167 9-8559

4"2131 39840 3-0443

1039"0 1032"1 1030.4

7.3674 6.7688 41794

3.2736 2.9601 1.2599

10

3-D PSD CST

3584 8 35733 35712

12.150 11.896 10.429

4.2457 4-1172 31239

2422-4 2410 0 24008

7-5123 7.0325 4.2784

3-2769 30347 1.2733

20

3-D PSD CST

7142.6 71236 7116-3

12"258 12.094 10583

42399 4.1518 3-1444

3632"2 3617"0 3596"5

7-5328 7.1016 4.3039

3.2669 30540 1.2766

CO

3-D PSD CST

12-257 12.161 10"636

4"1291 4"1635 3'1513

4356 9 4342-0 43125

75169 7.1250 4-3125

32525 3.0604 12777

10674 10651 10635

dimensional shell theories, when compared with the present three-dimensional elasticity solution, are considerable.

6. 7.

REFERENCES 8. 1 Snmvas, S., Rao, A. K & Joga Rao, C. V., Flexure of simply supported thick homogeneous and laminated rectangular plates. ZAMM, 49 (1969) 449-58. 2. Srlnivas, S. & Rao, A. K., Bending, vibration and buckhng of simply supported thick orthotropic rectangular plates and laminates, lnternattonal Journal of Sohds and Structures, 6 (1970)pp. 1463-81. 3 Greenspon, J. E., Flexural vibrataons of tluck walled circular cylinder. Proc U.S Nattonal Cong. Appl. Mech., Brown University, Providence, Rhode Island, 1958, pp 163-73. 4. Gazes, D. C., Three-dunenslonal investigation of the propagation of waves m hollow circular cylinders, I. Analytical Foundation. Journal of the Acousncal Soctety of Amertca, 31 (5)(1958) 568-73. 5. Chou, F,-H. & Achenbach, J. D., Three-dimensional wbratlons of orthotroplc cylinders. Journal of the Engt-

9. 10 11. 12. 13. 14. 15.

neenng Mechantcs Dtvtston, (ASCE), 98 (EM4) (1972) 813-22 Srinivas, S., Analysis of laminated, composite, circular cylindrical shells with general boundary condmons NASA TR, R-412, 1974. Flugge, W. & Kelkar, V. S., The problem of an elastic circular cylinder. Internanonal Journal of Sohds and Structures, 4 (1968) 397-420. Yao, J. C., Tubes under radial loads. Aeronautical Quarterly, 20 (1969) 365-81. Flugge, W., Stresses m Shells. Springer-Verlag, Berhn, 1962. Gol'denvelzer, A. L, Theory of Elastic Thm Shells. Pergamon Press, Oxford, 1961. Sanders, J. L., Jr, An improved first approximation theory for thm shells. NASA TR, R-24, 195-9. Timoshenko, S. & Womowsky-Krieger, S., Theory of Plates and Shells, McGraw-Hill Book Co, New York, 1959. Mindlin, R. D., Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates. Journal of Apphed Mechamcs (ASME), 73 ( 1951 ) 31-8. Mtrsky, I. & Herrmann, G., Nonaraally symmetric mouons of cylindrical shells. Journal of the Acousttcal Society of America, 29 (10) (1957) 1116-24. Reissner, E., The effect of transverse shear deformation

Alavandt Bhtmaraddt

76

18 Blumarad&, A. & Chandrashekhara, K., Three-dunensmnal elastloty solutmn for static response of s~mply supported cylindrical shells. Compostte Structures, 20 (4)(1992) 227-35 19 Saada, A S., Elasttctty Theory and Apphcaaons Pergamon Press, New York, 1974. 20 Bhlmaraddl, A., A higher-order theory for free vibranon analysis and circular cylmdncal shells lnternattonal Journal of Sohds and Structures, 20 (7) (1984) 623-30

on the bending of elastic plates. Journal of Apphed Mechantcs (ASME), 12 (1945)A-69 16 Bhlmarad&, A, Static and transmnt response of cylindrical shells Thm-Walled Structures, 5 (3) (1987) 157-79 17 Stem, M., Nonlmear theory for plates and shells including the effects of transverse shearing AIAA Journal, 24 (9)(1986) pp 1537-44

APPENDIX Definitions of c l-c7 appearing m eqns (12) and (14) are

c l = al6bl3;

c2 = a16b14 + a17b13

c 3 = 416b15 + a17b14 + a18b13 + al2b 1 + al4b 5 c a = a16b16 + a17b15 + a18b14 + al2b 2 + a13b I + a14b6 + a15b 5 c s = a16b17 + al7bl6 + a~8b1 s + al2b 3 + al3b 2 + 414b7 + a15b 6

c 6 = a17b17 + a18b16 + a12b4 + al3b 3 + a14b 8 + al5b 7 c 7 = a18b17 + al3b 4 +

a~5b8

Definitions for bl-b45 appearing in eqn (17) are b 2 = - a5a8 - a6a7;

b~ = - 45a7;

b 4 = a 4 a l l - a6a9;

b 3 = a4a m - asa 9 - a6a 8

bs--- - a t a m ;

b7-- -a3alo-a2all + a4a5; b9 = (Nb1+ Mbs)C66;

b12 = (Nb 4 + Mb8)C66;

b8 = -a3a11+a4a6

blo =(Nb2 + Mb6)C66; bl3 = ala7;

b15= a~a 9 + a2a 8+ a3a7; bib = (b 1 + Mb13)C44;

b6= - a ~ a ~ - a 2 a l o

bll =(Nb3 + Mb7)C66

b14 = ala 8 + a2a 7

b16 = a2a 9 + a3a8;

b17 = a3a 9 - a 2

b19=(b2+Mb14-bl/R1)Ca4,

b20=(b3+Mbls-b2/R1)C44

b21 = (b4 + Mbl6 - b3/R 1)C44;

b 2 2 = ( M b 1 7 - ba/R1)C44;

b24 = (b 6 + Nb14 - bs/R2)C55;

b25 = (b 7 + Nb15 - b6/R2)C55

b26 = ( b 8 + Nbl6 - b7/R2)C55;

b27 = (Nbl7 - bs/R2)Cs5;

b29=[R1

R 2 J bla+Clab14;

b31

JR1

R2J

b32

[ R1

RE J

b33

JR1

REJ

b35

[ gl

RE j b13 + C2sb14;

b30

[ R1

b23=(bs + Nb13)C55

b2~ = blaCj3

R2 J b14 + C13b15 - CliMb1 - C12Nbs

b15 + C13b16- C l i M b 2 - CI2Nb6

b16 + C13b17- C l i M b 3 - C12Nb7

b i t - C l i M b 4 - C12Nb8 ",

b36

b34 = C23b13

=Ic12+c2,1 --RTJ bl'+CEab''-C~EMb~-CE2Nb' [_-~1

77

Analysts of orthotroptc doubly curved shallow shells

bay= C,2+ C22] R1

R2 j bl5 + C23b16 -

C l g M b 2 - C22Nb6

R2 ] b16+ C23b1~ -

C12Mb3 - C22Nb7

R2 .] b17 -

C l z M b 4 - C22Nb8;

b41

[ R1

R2 J b13 + C33b14;

b43

[_R1

R2 _]ba5+

Cl3 + C23]

b44 = ~

C33b6 -

~-2 .] hi6 + C33b17-

b42

[ R1

b4o=

R2 _] b14 + C33b15 - C13Mbl

C13Mb2 - C23Nb9

C 1 3 M b 3 - C23Nb7

_[C13..t_~-2J C23] b17-f13Mb4-f23Nb8 b45~..~LRI

C33b13

- C23Nb5