Effects of general laminations and boundary constraints on vibration of composite shallow shells

ELSEVIER 1359-8368(95)00038-0

Composites: Part B 27B (1996) 155 171 Copyright © 1996 Elsevier Science Limited Printed in Great Britain. All rights reserved

1359-8368/96/$15.00

Effects of general laminations and boundary constraints on vibration of composite shallow shells

K. M. Liew Division of Engineering Mechanics, School of Mechanical and Production Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798

and C. W. Lim and S. Kit ipornchai Department of Civil Engineering, The University of Queensland, Brisbane, Queensland 4072, Australia (Received 11 May 1995; accepted 10 October 1995)

This paper presents a comprehensive parametric study on the essential effects of symmetric and unsymmetric fibre orientations on vibration of thin, composite double curved (cylindrical, spherical and hyperbolic paraboloidal) shallow shells with arbitrary and complex boundary constraints. The precise strain and kinetic energy integral expressions for the laminates are formulated by using the extremum energy principle with no restrictions on the number of laminations and fibre orientations. A versatile and flexible general method of solution for vibration of composite laminates is developed. Convergence and comparison studies are reported to verify the consistency and accuracy of this analysis method. New results are presented for simply supported and fully clamped laminated shells with multiple laminations, symmetric and unsymmetric stacking sequences for two types of composite materials. The influence of fibre angles and stacking sequences on vibration mode shapes is also illustrated.

(Keywords: boundary constraint; laminated composite; mode shapes; Ritz method; shallow shells; vibration frequency)

1 I N T R O D U C T I O N

Composites are the most promising materials for components of current and future engineering structures, with significant demand at present in the aircraft and aerospace industries. They have many structural advan- tages over isotropic materials, including great potential for weight saving due to their high strength-to-weight ratios. The structural properties of composite materials can be adjusted to design requirements by tuning the matrix, lamination and fibre combination and orientation.

Although much has been published on the vibration of composite plates I 10 reports on composite shells deal

ma in ly with closed cylindrical and spherical shells lk12 because of their practical importance in aerospace, petro- chemical and nuclear industries. Bert I 4, in his excellent publications, has listed a large number of references in dynamic analysis of composites and sandwich panels. Most of the studies are confined to plates of rectangular domain 5'6. A limited number of references are available

for laminates of other shapes, such as Liew and Lim 7 in which the vibration of generally laminated trapezoidal plates was investigated. The effects of pretwist on natural frequency have also been studied by Liew and Lim s in an effort to simulate the vibration of turbomachinery blades. Recently, Mirza l~ surveyed recent research on the vibration of laminated shells, assessing works published on free, forced vibration and dynamic response of layered shells. Mirza's review covers studies of analytical tech- niques, finite element analyses and experimental methods. Nonlinear and large deformation effects were also presented.

Few investigations into the free vibration of composite shallow shells have been undertaken. One early work in this field was reported by Crawley 13 who investigated the natural vibration of graphite/epoxy cantilevered shallow shells. Some finite element solutions and experimental results were reported. Recently, an exhaustive study of the equation of motion, boundary conditions and energy functional was presented by

155

Vibration of composite shallow shells: K. M. Liew et al.

Qatu 14 and Leissa and Qatu Is. The latter publication presented a complete and consistent elastic deformation theory for static deflections, stresses, free and forced vibrations. Exact and approximate solutions (Galerkin, finite difference, Ritz, etc.) can be obtained from these equations. Employing the Ritz method with simple polynomial functions, a complete set of vibration frequencies for cantilevered doubly curved shallow shells was reported by Qatu and Leissa 16, while another parametric study on the completely free shells was also reported 17. The formulation and method of solution 14'16'17 can only be used to solve composite plates and shells with simple boundary conditions although the theory encom- passes shells with general lamination (symmetric and unsymmetric), including extension-shearing, bending- stretching and bending-twisting couplings. However, no result was presented for unsymmetrically laminated shells, i.e. only shells without bending-stretching couplings (Bij = 0) were examined and thus the computation can be simplified greatly. Laminates are seldom restricted to only symmetric stacking sequences with respect to the mid-surface because unsymmetric laminations provide an additional choice for engineers to tune the mechanical properties to suit design requirements. Therefore, the availability of numerical results for unsymmetrically laminated shallow shells is essential. Furthermore, the analyses 14'16J7 are also confined to completely free and cantilevered shells although other boundary constraints are also of practical importance. The completely free shallow shells have little use in normal practice, but perhaps are useful in the design of aerospace structures. Although cantilevered shallow shells form important structural components, other types of boundary conditions, such as simply supported and fully clamped boundaries, are also equally important, particularly in the design of buildings and large civil structures.

To fill this apparent gap, Liew and Lim 7'8 presented computational results for generally laminated.trapezoidal plates with and without the effect of pretwist. Research presented herein has been undertaken to extend and generalize previous studies 7'8'14'16'17 and to provide results for generally laminated shells with arbitrary boundary conditions. The derivation of energy integral equations is presented using the Ritz extremum energy principle and the eigen-matrix equation is constructed. Two kinds of composite materials are investigated, namely E-glass/ epoxy and graphite/epoxy. Unlike the analyses by Qatu 14 and Qatu and Leissa 16J7, this study has considered complex and arbitrary boundary conditions and described a class of kinematically oriented, orthogonally generated global shape functions. These admissible shape functions incorporate the boundary conditions intrinsically and are extremely flexible and versatile. As a result, various combinations of boundary conditions can be easily manipulated and accurate approximate solutions obtained. For descriptive purposes, the simply supported and fully clamped laminated shallow shells are analysed in this paper where no results are currently available. The influences of various parameters such as angle of

lamination, number of laminae, stacking sequence and shallowness and curvature ratios are also discussed at length. Furthermore, to illustrate the effects of these parameters, contour mode shapes are presented to enhance vibration visualization.

2 THEORY AND ANALYSIS

Consider a thin, doubly curved fibre reinforced laminated composite shell as shown in Figure 1. The shell is composed of generally laminated plies parallel to the mid-surface. The angle of fibre orientation is /3 and overall thickness is h with all laminae having equal thickness. The shell is of rectangular planform with length a, width b and radii of curvature Rx and Ry as shown in Figure 1. The planform is bounded by -a /2 < x < a/2 and -b /2 <_ y <_ b/2 and subjected to a combination of free, simply supported and clamped boundaries. The deflections are resolved into three orthogonal components u, v and w with respect to the mid-surface of the shell with u and v tangential to the mid-surface (u parallel to the xz-plane and v parallel to the yz-plane) and w normal to it.

The strain energy, 0g, of a laminated shell can be determined by integrating over the volume of the laminael8:

I I I l (ffx+~x+ffy£y+~rxy'3'xy)dV. (1) v

The stress-strain relationships of the kth lamina can be expressed by the transformed reduced stiffness (0q)k in the following constitutive relation:

{ xlk ro,, °,2 _161 IQl2 022 0_26 / (2)

Crxy [QI6 Q26 O66]k "Yxy k

where (Qij)k is a function of the stiffness constants (Q/j)k and the fibre orientation angle 13 k given by18:

~)llk = Qllk c4 + 2(Q12k + 2Q66k) s2c2 + Q22k $4 (3a)

T / /

' 2

~ y R 1 \ / '\

Figure 1 Geometry of a doubly curved composite shallow shell

156

Vibration of composite shallow shells. K. M. Liew e t a l .

012k = (Qllk + Q22k - - 4Q66k) $2c2 q'- Q12k( $4 if- C4) (3b)

016k = ( Q l l k - - Q12k -- 2Q66k) Sc3

+ (Q12k - - O22k -t- 2Q66k)SBc (3c)

022, = Qllk s4 --}'- 2(Ql2k + 2Q66k) s2c2 q- Q22k c4 (3d)

026k = (Qll, -- Ql2k -- 2Q66k) s3c

+ (Qlzk -- Q22k + 2Q66~) sc3 (3e)

Q66k = (Qlk - 2QlZk -4- Qz2k -- 2Q66k) s2c2

q- Q66, (s 4 q- c 4)

where c and s denote cos

(3f)

flk and sin ilk, respectively, and

Qllk-- " Ell, (4a) 1 - - /,/12k//21k

-- P12kE22k (4b) Q12~ 1 -- /)12k//21k

Q22k -- E22k (4C) 1 - - /]12kb'21k

Q66k = Gl2k (4d)

t"21kEllk = t/12kE22k (4e)

in which Ell, and E22 k a r e , respectively, the Young's moduli parallel and perpendicular to the fibres; u~2 k and //21 k a r e the corresponding Poisson's ratios and Gl2 is the shear modulus.

Substituting equation (2) into equation (1) and using the strain-displacement relationships to integrate through the shell thickness,

+ o; I x}o w "~ Ou Ov 20zw

the strain integral (1) can be reduced to:

~ ' = d~tor q- ~ffes q- 4]/bs ( 6 )

where q/or = terms due to uncoupled orthotropic characteristics of the material (i.e. All, d12, A22, A66, Dll , D12, D22 and D66); Y//es = terms of extension-shearing coupling (i.e. At6 and A26); and q/bs = terms of bending- stretching coupling (i.e. B! j ) 14'16'17. T h e s e laminate stiffness coefficients Aii , B~y and Dij arel8:

n

Aij = ~_,(O_ij)k(hk--hk-1); ( i , j = 1,2,6) (7a) k=l

l n _

B!i= ~ Z ( Q i j ) k ( h 2 - h 2 1); ( i , j = 1,2,6) (7b)

1 ~ h 3 3 = - h k ]); ( i , j= ,2,6) (7c) Dij ~ (Qij)k( k 1 k l

The bending-stretching coupling vanishes (B/j = 0) for

symmetrically laminated shells. The computational aspects of this coup'ling, not found in Qatu 14 and Qatu and Leissa 16'17, are investigated in this paper.

The kinetic energy for free vibration is given by

[(0.) 2+ /ov 2 /Ow 21 dA (8)

where p is the mass density per unit volume. Assuming periodic displacement functions in time of

the following forms

u(x,y, t, = U(x,y) sincot (9a)

v(x, y, t) = V(x, y) sin aJt (9b)

w(x,y, t) = W(x,y) sincvt (9c)

where U(x,y), V(x,y) , W(x ,y) are the displacement amplitude functions and w denotes the frequency of vibration.

The maximum strain energy 0"#ma x and maximum kinetic energy J-max in vibratory cycle, which occur at maximum and zero displacements, respectively, can be derived by substituting equation (9a-c) into equations (6) and (8). Following the extremum energy principle, the energy functional

~ = d~'max - ~-ma x (10)

is minimized in accordance with the Ritz method as described in the following section.

3 M E T H O D OF SOLUTION

3.1 Global shape functions

Introducing a non-dimensional coordinate system

_x. y (11 a, b) ~ = a ' ~?=b

where a and b are the span and width of the shell as shown in Figure 1, the in-plane and transverse displacement amplitude functions can be approximated by a series of orthogonally generated two-dimensional polynomials

U(Gq) = ~ C~'qS}'(~, 71) (lZa) i=1

V(~, r/) = ~ C ~ ( ~ , r/) (lZb) i=1

m(~, q) = ~ C~"O2'(~, q) (12c) i=1

where C}', Cy and Ci w are the unknown coefficients and qS~', 4~: and ~Wwi are the corresponding pb-2 shape functions. These functions are admissible functions consisting of the product of terms of a two-dimensional orthogonally generated polynomial (p-2) and appropriate basic functions (b), i.e.

i 1 ct ~ct (x ~i (~,~]) =f i (~ ,7~)~ - - Z - - ( j ~ j ( 1 3 )

j = l

157

Vibration of composite shallow shells. K. M. Liew et al.

where

=o~ lmu (14a) ~ 0 " - 2A 7

(14b)

2/k;' = I Ia(qS~)2d{dr/ (14c)

in which a = u, v or w. ~imlf/(~, q) forms a complete set of p-2 functions and 0~ (a = u, v or w) are the basic functions defined by the products of the equations of the continuous piecewise boundary geometries of the shell planform each of which is raised to an appropriate power that corresponds to boundary conditions.

In the present approach, the basic functions for different boundary conditions are given by:

he

= II[r,(e, i=1 ,{o

O'ffi = 1

He

i=1 {0 "7~ = 1

Be

" HI O~ (~, v) = r~(~, ~ ) 7 r i=1

{ ° o7~' = 1

2

15a)

free;

u-displacement constrained;

(15b)

free;

v-displacement constrained;

(15c)

free;

simply supported;

clamped;

where n¢ is the number of supporting edges and F i is the boundary expression of the ith supporting edge.

The two-dimensional polynomial ~m=lJ}(~ , ~7) can be expressed as

m p q

E fi(~'7]) = E E ~q iI]i (16) i=1 q=o i=0

with m and p related by

(pqZ l)(p 4- 2) m = 2 (17)

where p is the degree of the complete set of two- dimensional polynomials employed.

3.2 Governing eigenvalue equation

The minimization of equation (10) with respect to the unknown coefficients C~', C~ and C~" leads to the following governing eigenvalue equation:

(K - A2M){C} = {0} (18)

where K is the stiffness matrix and M is the mass matrix

expressed as follows:

K =

M =

kU. k~ k ~ ' ]

k vv k vw

sym k ww

- m "~ Io] [o]

m v~ [0]

sym m w"

(19a)

(19b)

and the vector of unknown coefficients is {,,} { C } = {C} . (19c)

{c ~ }

The elements in the stiffness and mass matrices are:

k~y _ b2All ¢1010 abA16 (¢0!19 ¢1001] D~-=uiu) 4- Do ~u~uj 4- ~uiuj j

a2A66 ¢0!0.1 (20a) 4- Do ~utuj

uv _ a2A26 ¢0!0.1 ko " abAl2 jlOql , b2A16 .~1010 4_ Do --uz~ Do u,~j ~- O~- ~uiv/

4- abA66 ¢0!1o (20b) Do --uw

ab2A11 ¢100o , ab2At2 jlooo , a2bA16 jmoo k~jW- RxDo vuiwj ± RyDo ui.j + ~ ui.j

a2bA26 ¢oloo b2Bll ¢1920 aBj2 ¢1002 RyD~ --uiwj aDo vu,,~,j Do --.iwj

bB16 (j0!20 4- 9¢1011~ _ a2B26 ¢0!02 Do ~ u,wg -- --~uiwj J bDo ~u,,,:l

2aB66 .¢0111 (20c) - D----~ --uiwj

vv a2A22 ¢0101 abA26 {¢0110 ¢1001 kij - - Do =vivj 4- - ~ o ~vivj 4- ~VtVl

b2A66 .¢1010 4- ~ ~vivj (20d)

k~' - a2bA12 jOlOO ab2A16 .¢1909 a2bA22 jOlOO Rx Do ~i.j 4- Rx Do ww 4- Ry D~-~- viwj

ab2A26 jlOO 0 bB12 jo!20 b2Blh jm20 , ~ viwj" 4- Ry Do wwj Do wwj aLPo

a2B22 jo102 aB26 (¢1902 , ,~ .~Olllx bD~--~ vi,j Do ~wwj ± z~vi,j )

2bB66 ¢191! (20e) Do ~v,w

a2b2All joooo ~ j o o o o ~ ~ ,,,.,J kijWW - R2xD-----~ wi,~j + a2bEA22 joooo x y o =-y ~o

b2 Bl l (jo020 ¢200% _ a2 B12 G¢0002 j0200~ RxDo ~ w.v 4- ~wt),y J R ~ o ° ~ wiuj 4- winj j

158


Table 1 Convergence of A = w a b ( x / ~ / D o ) for the SSSS 8-ply laminated E-glass/epoxy shells (E2/EI = 0.409, GI2/E 1 = 0.198, ul2 = 0.23) with a/h = 100 and a/b = 1.0

p Mode sequence number

SS u v w 1 2 3 4 5 6 7 8

[(-30 ~, 30"~)4]sym -

[(-30, 30°)4]unsym

6 6 6 70.835 74,617 108.16 109.66 115.66 126.68 129.49 139.47 6 6 7 70.811 74.617 108.16 109.17 115.66 122.90 126.98 139.47 6 6 8 70.811 74,612 108.12 109.17 115.62 122.90 126.98 138.42 6 6 9 70.811 74.612 108.12 109.16 115.62 122.80 126.92 139.42 6 6 10 70.811 74,612 108.12 109.16 115.62 122.80 126.92 139.42 6 7 10 70.811 74.562 107.98 109.16 115.50 122.80 126.92 139.37 6 8 10 70.809 74.562 107.98 109.14 115.50 122.61 126.80 139.37 6 9 10 70.809 74.561 107.98 109.14 115.50 122.61 126.80 139.37 6 10 10 70.809 74.561 107.98 109.14 115.50 122.61 126.79 139.37 7 10 10 70.809 74.555 107.98 109.14 115.49 122.61 126.79 139.35 8 10 10 70.809 74.555 107.98 109.13 115.49 122.60 126.79 139.35 9 lO 10 70.809 74.555 107.97 109.13 115.49 122.60 126.79 139.35

10 1 0 lO 70.809 74.555 107.97 109.13 115.49 122.60 126.79 139.35

6 6 6 70.823 74.631 108.18 109.73 115.66 127.69 128.43 139.39 6 6 7 70.800 74.631 108.18 109.29 115.66 123.39 126.52 139.39 6 6 8 70.800 74.627 108.15 109.29 115.63 123.39 126.52 139.34 6 6 9 70.800 74.627 108.15 109.28 115.63 123.29 126.47 139.34 6 6 10 70.800 74.626 108.15 109.28 115.63 123.29 126.47 139.34 6 7 10 70.800 74.578 108.02 109.28 115.52 123.29 126.47 139.30 6 8 10 70.798 74.578 108.02 109.26 115.52 123.10 126.36 139.30 6 9 10 70.798 74.578 108.02 109.26 115.52 123.10 126.36 139.30 6 lO 10 70.798 74.578 108.02 109.26 115.52 123.10 126.36 139.30 7 10 10 70.798 74.572 108.01 109.26 115.51 123.10 126.36 139.28 8 10 10 70.798 74.572 108.01 109.25 115.51 123.08 126.36 139.28 9 10 I0 70.798 74.572 108.01 109.25 115.51 123.08 126.36 139.28

10 10 10 70.798 74.572 109.01 109.25 115.51 123.08 126.35 139.28

SS: Stacking sequence of laminates

Table 2 Convergence of A = ~vab(x/~/D o ) for the CCCC 8-ply laminated E-glass/epoxy shells (Ez/E 1 = 0.409, G12/E 1 = 0.198, u12 = 0.23) with a/h - 100 and a/b = 1.0

p Mode sequence number

SS u v w 1 2 3 4 5 6 7 8

[(--30 c, 30°)4]sym

[(-30 °, 30°)4]unsym

6 6 6 77.557 87.441 119.53 119.87 121.40 144.61 154.07 157.84 6 6 7 77.551 87.441 119.53 119.87 121.40 144.42 153.28 157.84 6 6 8 77.551 87.438 119.53 119.87 121.39 144.42 153.28 157.83 6 6 9 77.551 87.438 119.53 119.87 121.39 144.41 153.26 157.83 6 6 10 77.551 87.437 119.53 119.86 121.39 144.41 153.26 157.83 6 7 10 77.551 87.302 119.47 119.53 121.02 144.41 153.26 157.70 6 8 10 77.545 87.302 119.47 119.50 121.02 143.83 152.55 157.70 6 9 10 77.545 87.301 119.46 119.50 121.01 143.83 152.55 157.70 6 10 10 77.545 87.301 119.46 119.50 121.01 143.82 152.53 157.70 7 10 10 77.545 87.284 119.43 119.50 120.99 143.82 152.53 157.68 8 10 10 77.545 87.284 119.43 119.50 120.99 143.77 152.59 157.68 9 10 10 77.545 87.284 119.43 119.50 120.98 143.77 152.49 157.67

10 10 10 77.545 87.284 119.43 119.50 120.98 143.76 152.48 157.67

6 6 6 77.554 87.454 119.53 120.08 121.18 145.18 153.57 157.65 6 6 7 77.548 87.454 119.52 120.08 121.18 145.02 152.78 157.65 6 6 8 77.548 87.452 119.52 120.07 121.17 145.02 152.78 157.64 6 6 9 77.547 87.452 119.52 120.07 i21.17 145.01 152.75 157.64 6 6 10 77.547 87.451 119.52 120.07 121.17 145.01 152.75 157.64 6 7 10 77.547 87.320 119.52 119.72 120.78 145.01 152.75 157.52 6 8 I0 77.542 87.320 119.50 119.72 120.78 144.48 152.03 157.52 6 9 10 77.542 87.319 119.50 119.71 120.77 144.48 152.03 157.52 6 10 10 77.542 87.319 119.50 119.71 120.77 144.48 152.01 157.52 7 10 10 77.542 87.302 119.50 119.69 120.74 144.48 152.01 157.49 8 10 10 77.542 87.302 119.50 119.69 120.74 144.43 151.96 157.49 9 10 10 77.542 87.302 119.50 119.69 120.74 144.43 151.96 157.49

10 10 10 77.542 87.302 119.50 119.69 120.74 144.43 151.96 157.48


159

Vibration of composite shaflow shells." K. M. Liew et al.

Table 3 C~mp4aris~n ~f ana~ytica~ experimenta~ and c~mputati~na~ natura~ frequencies (Hz) f~r canti~evered p~ates with a = ~.3~48 m~ b = ~.~762 m , h = 8.04 × 10- m, p = 1520kg/m , u = 0.28, E n = 98GPa, E22 = 7.9GPa and GI2 = 5A6GPa

Jensen el al. 6

Laminations MS FEM A B C D A' D' Exp QL pb-2

[0 °, 0 °, 90°]sym 1 11.1 11.1 11.1 11.1 11.1 11.1 11.1 11.2 11.04 11.20 2 39.5 39.6 39.6 39.6 39.6 40.0 40.0 42.4 39.49 38.25 3 69.5 69.4 69.4 69.3 69.3 69.4 69.3 70.5 69.16 69.25

[15 °, 15 °, Oe]sym 1 8.9 8.8 8.8 8.8 8.7 9.0 9.0 9.4 8.923 8.830 2 42.9 50,0 48.8 49.4 48.2 46.5 44.0 45.8 42.82 42.05 3 62.7 67.7 60.5 66.5 59.9 70.3 65.0 66.2 62.52 62.23

[30 °, 30 °, O°]sym 1 6.3 6.3 6.2 6.3 6.2 6.5 6.4 6.6 6.288 6.181 2 37.3 53.3 42.0 52.5 42.0 49.1 41.0 40.0 37.22 36.64 3 56.9 69.9 69.0 60.8 60.7 73.0 59.7 59.1 56.87 56.59

[45 °, 45 °, 0°]sym 1 4.9 4.9 4.9 4.9 4.8 5.0 5.0 4.8 4.875 4.804 2 30.1 41.4 32.7 38.5 32.6 39.9 33.0 29.8 30.02 29.63 3 49.4 74.0 73.9 59.1 56.3 74.9 52.4 51.3 49.55 48.86

[60 °, 60 °, Oe]sym 1 4.2 4.2 4.2 4.2 4.2 4.3 4.2 4.3 4.179 4.144 2 26.1 30.4 27.0 28.7 26.8 29.9 27.0 27.1 26.04 25.84 3 41.7 65.4 65.4 50.4 47.1 65.6 44.1 47.7 41.81 40.72

[75 °, 75 °, 0°]sym l 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.8 3.877 3.869 2 24.3 24.9 24.5 24.7 24.4 24.8 24.4 25.1 24.27 24.23 3 36.7 47.2 47.2 40.1 39.2 47.3 37.9 38.9 36.78 35.52

[90 °, 90 °, 0°]sym 1 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.7 3.801 3.806 2 23.9 23.9 23.9 23.9 23.9 23.9 23.9 24.3 23.82 23.85 3 35.1 35.1 35.1 35.1 35.1 35.2 35.2 38.2 35.07 33.79

MS: Mode sequence QL: Qatu and Leissa 16

Table 4 Co_m3Parison of analytical, experimental and computational natural frequencies (Hz) for cantilevered plates with h = 1.04mm, p = 1500kg/m , u = 0.25, Ell = 128 GPa, E22 = 11GPa and Gl2 = 4.48 GPa

Mode sequence number

a = 0.0762m, b = 0 . 0 7 6 2 m a = 0.1524m, b = 0.0762m

Laminations References 1 2 3 4 5 1 2 3 4 5

[0 °, 0 °, 30 °, - 30]syrn FEM 13 261.9 263.0 761.8 1662 1709 65.37 1 3 7 . 5 408.3 525.6 588.3 Experimen03 234.2 262.0 728.3 1449 1503 58.30 148.0 362.7 508.0 546.0 Qatu and Leissa 16 260.3 362.1 757.6 1596 1632 64.93 136.9 405.8 523.0 579.9 Present pb-2 method 260.3 361.8 756.1 1592 1632 64.93 136.8 405.8 522.5 578.6

[0 °, 45 °, 45 °, 90°]sym FEM 13 224.3 421.8 1012 1426 1722 55.58 175.4 345.3 591.8 820.1 Experiment 13 196.4 418.0 960.0 1215 1550 48.6 169.0 303.0 554.0 739.0 Qatu and Leissa 16 222,9 421.2 1010 1405 1705 55.24 175.0 343.4 591.0 808.8 Present pb-2 method 222.8 421.0 1009 1405 1704 55.22 174.9 343.3 590.6 808.0

[45 °, -45 °, 45 °, 45°]sym FEM 13 138.9 499.5 805.0 1326 1648 31.90 1 9 1 . 3 228.2 565.3 708.3 Experiment 13 131.2 472.0 790.5 1168 1486 31.3 185.8 214.0 533.0 653.0 Qatu and Leissa 16 138.5 500.0 801.2 1319 1640 31.99 191.6 228.7 564.8 708.8 Present pb-2 method 138.3 499.6 799.3 1318 1639 31.95 191.4 228.7 564.3 708.6

Table 5 Comparison of analytical, experimental and computational natural frequencies (Hz) for cantilevered cylindrical shells with a = 0.1524 mm, b = 0.0762mm, h = 1.04mm, p = 1500kg/m 3, Ry = 0.1275m, u = 0.25, E n = 128 GPa, E22 = 11GPa and Gl2 = 4.48 GPa


Laminations References 1 2 3 4 5 6

[0 °, 0 °, 30 °, -30°]sym FEM 13 165.7 289.6 597.0 718.5 833.3 Experiment 13 161.0 254.1 555.6 670.0 794.0 Qatu and Leissa 16 163.8 269.5 593.0 702.5 824.8 Present pb-2 method 166.2 282.4 603.9 713.5 832.1

[0 °, 45 °, -45 °, 90°]sym FEM 13 192.4 236.1 705.8 808.2 980.1 Experiment 13 177.0 201.8 645.0 754.0 884.0 Qatu and Leissa 16 190.8 217.5 695.5 804.8 958.1 Present pb-2 method 193.2 229.6 705.9 815.8 977.1

[45 °, -45 °, -45 °, 45°]sym FEM 13 147.0 238.0 768.1 812.1 - - Experiment 13 145.3 222.0 712.0 774.2 Qatu and Leissa 16 133.1 236.2 746.6 758.8 842.1 Present pb-2 method 140.1 238.6 765.7 780.4 - -

m

r

m

1038 997.0

1037 1054

160


Table 6 Comparison of the frequency parameters wa2x/p/Ellh 2 for completely free, angle-ply, laminated shallow shells (graphite/epoxy) with a/b = 1.0, a/h = 100 and stacking sequence [45 °, - 4 5 °, 45°]sym or [ -45 °, 45 °, -45°]sym

g x a

Ry Rx References


1 2 3 4 5 6 7 8 9 10

- 1 0.1 Qatu and Leissa 17 1.6901 3.8269 4.2699 4.7081 5.9158 7.2423 9.1294 9.3454 11.361 12.277 Present pb-2 method 1.6901 3.8265 4.2697 4.7081 5.9113 7.2324 9.1168 9.2979 11.327 12.051

0.5 Qatu and Leissa 17 1.6914 3.8431 5.0012 7.3139 9.5905 10.674 14.847 17.967 18.240 18.850 Present pb-2 method 1.6907 3.8417 4.9930 7.3077 9.5830 10.578 14.502 17.902 18.200 18.459

0 0.1 Qatu and Leissa 17 1.7815 2.8868 3.9459 4.8734 5.4328 7.5553 8.8658 8.9693 11.227 12.317 Present pb-2 method 1.7814 2.8867 3.9455 4.8722 5.4285 7.5439 8.8545 8.9234 11.187 12.084

0.5 Qatu and Leissa 17 1.8141 4.0935 4.6058 7.1171 7.5336 10.026 10.753 13.761 14.014 15.170 Present pb-2 method 1.8140 4.0913 4.6035 7.1079 7.5124 10.007 10.653 13.700 13.860 14.584

1 0.1 Qatu and Leissa 17 2.2445 2.2882 4.3005 5.2419 5.3758 8.4880 8.9032 8.9316 11.355 12.975 Present pb-2 method 2.2444 2.2881 4.3002 5.2406 5.3731 8.4773 8.8649 8.9201 11.306 12.748

0.5 Qatu and Leissa 17 2.2761 2.9634 5.9725 6.3364 8.2020 10.206 12.033 13.681 15.681 19.372 Present pb-2 method ' 2.2747 2.9605 5.9673 6.3310 8.8190 10.184 11.975 13.618 15.573 18.624

Table 7 Frequency parameters A = ~ a b ~ f o r SSSS singly curved and doubly curved, 4-ply laminated E-glass/epoxy shells with a/h = 100.0 and a/b = 1.0

Mode sequence number b R y

SS Ry R x fl 1 2 3 4 5 6 7 8

( - f l , fl, f l , - f l ) 0.0 all 0 ° 15.195 33.300 44.419 60.779 64.530 90.302 93.664 108.56 15 ° 15.462 34.200 43.897 61.257 66.558 91.456 92.155 109.11 30 ° 15.994 36.268 42.588 62.482 71.634 85.774 95.673 109.18 45 ° 16.259 37.662 41.575 63.172 77.500 79.534 97.379 109.19

0.5 -1 .0 0 ° 110.94 118.10 124.53 126.23 147.48 151.42 175.00 179.69 15 ° 111.63 118.66 125.17 127.28 148.74 151.82 170.11 173.17 30 ° 113.00 118.15 124.83 128.18 143.38 149.43 157.90 158.75 45 ° 114.41 117.40 122.77 128.06 144.84 145.36 150.88 153.11

0.0 0 ° 62.301 67.212 98.183 99.702 107.53 112.18 118.54 138.94 15 ° 65.011 69.231 100.96 102,36 109.08 114.62 120.35 138.70 30 ° 7.0.599 74.165 107.33 108,20 115.37 121.07 127.22 139.04 45 ° 75.363 80.140 114.23 114,65 128.35 131.57 139.04 143.89 60 ° 76.596 85.821 117.69 117,82 145.60 146.46 151.93 152.27 75 ° 74.245 90.103 115.60 117.21 151.28 152.97 159.80 160.40 90 ° 72.399 91.774 113.06 116,37 148.81 150.91 164.96 166.22

1.0 0 ° 131.82 142.83 153.96 155.72 174.22 175.78 187.70 196.14 15 ° 135.25 143.28 158.39 159.03 173.67 179.76 185.23 195.89 30 ° 143.71 145.68 161.04 165.63 168.14 180.06 185.06 190.60 45 ° 150.20 153.05 155.25 157.39 171.49 181.19 183.25 187.61

( - ~ , f l , - ~ , ~ ) 0.0 all 15 ° 15.502 34.344 43.788 61.922 66.182 91.107 93.057 108.90 30 ° 16.118 36.890 42.147 64.225 70.848 84.964 99.267 107.95 45 ° 16.425 39.758 39.758 65.382 77.539 77.469 104.95 104.95

0.5 -1 .0 15 ° 112.08 119.86 125.06 126.12 148.77 152.14 170.44 173.86 30 ° 114.49 121.32 124.12 125.30 149.51 149.75 157.92 159.32 45 ° 118.95 118.95 121.35 124.57 145.15 150.16 150.16 151.13

0.0 15 ° 64.969 69.247 100.98 102.52 109.10 114.80 120.20 138.51 30 ° 70.557 74.236 107.49 108.73 115.43 122.47 126.03 138.75 45 ? 75.371 80.217 114.47 115.68 128.47 134.11 136.48 143.70 60 ° 76.612 85.804 117.79 118.86 145.77 147.98 151.10 152.49 75 ° 74.240 90.058 115.59 117.59 151.24 153.38 159.98 160.51

1.0 15 ° 135.17 143.07 158.54 159.1l 173.61 179.80 185.31 195.16 30 ° 143.58 145.45 161.15 166.02 167.77 181.19 185.63 189.24 45 ° 150.12 153.08 155.00 157.30 171.58 183.51 184.72 186.08


161


Table 8 Frequency parameters A = ~abphx/ph~ o for SSSS singly curved and doubly curved, 8-ply laminated E-glass/epoxy shells with a/h = 100.0 and a/b = 1.0

Mode sequence number b R v

SS Ry R~ /3 1 2 3 4 5 6 7 8

( - 3 , 3, - 3 , 3, 3, -3, /3, -/3) 0.0 all

0.5 -1 .0

( - 3 , 3, - 3 , 3, - 3 , fl, - 3 , 3) 0.0 all

0.0

1.0

0.5 -1 .0

0.0

1.0

0 ° 15.195 33.300 44.419 60.779 64.530 90.302 93.664 108.56 15 ° 15.527 34.377 43.959 61.947 66.384 91.554 93.107 109.36 30 ° 16.175 36.945 42.599 64.276 71.458 85.929 99.025 109.38 45 ° 16.490 39.098 41.038 65.446 78.262 78.821 102.59 108.68

0 ° 110.94 118.10 1 2 4 . 5 3 126.23 147.48 151.42 175.90 179.69 15 ° 112.13 119.85 125.29 1 2 6 . 5 5 148.95 152.30 170.56 173.85 30 ° 114.44 120.86 124.73 126.59 149.74 150.05 158.47 159.44 45 ° 117.21 120.45 121.25 126.32 145.47 148.62 1 5 1 . 3 3 152.68

0 ° 62.301 67.212 98.183 99.702 107.53 112.18 118.54 138.94 15 ° 65.077 69.344 1 0 1 . 2 1 102.73 109.11 114.89 1 2 0 . 3 5 138.86 30 ° 70.809 74.555 107.97 1 0 9 . 1 3 115.49 122.60 126.79 139.35 45 ° 75.667 80.793 114.97 1 1 6 . 1 1 128.59 134.11 138.12 144.19 60 ° 76.869 86.460 118.18 119.29 145.90 148.61 152.16 152.81 75 ° 74.349 90.380 115.74 117.79 151.46 153.54 160.56 160.62 90 ° 72.300 01.774 113.06 116.37 148.81 150.91 164.96 166.22

0 ° 131.82 142.83 153.96 155.72 174.22 175.78 187.70 196.14 15 ° 135.33 143.47 158.66 159.14 173.71 180.07 185.66 195.85 30 ° 143.91 146.05 1 6 1 . 3 1 1 6 6 . 1 8 168.14 182.16 185.88 190.78 45 ° 150.55 153.80 1 5 4 . 8 9 157.77 171.97 184.11 184.72 188.08

15 ° 15.537 34.413 43.932 62.127 66.276 91.467 93.337 109.29 30 ° 16.205 37.105 42.482 64.758 71.209 85.726 1 0 0 . 0 5 108.92 45 ° 16.531 40.100 40.100 66.043 78.268 78.296 106.04 106.04

15 ° 112.24 120.19 125.26 1 2 6 . 2 1 148.96 152.38 170.64 174.02 30 ° 114.82 122.01 124.54 125.52 150.05 150.12 158.44 159.55 45 ° 119.32 119.32 121.88 124.99 145.54 150.87 150.87 151.37

15 ° 65.007 69.348 1 0 1 . 2 1 102.77 109.12 114.93 120.31 138.81 30 ° 70.798 74.572 1 0 8 . 0 1 109.25 115.51 123.08 126.35 139.28 45 ° 75.678 80.810 115.02 116.35 128.62 135.49 136.72 144.14 60 ° 76.872 86.455 118.20 119.54 145.94 149.52 151.28 152.84 75 ° 74.348 90.369 115.74 117.89 151.45 153.64 160.59 160.70

15 ° 135.30 143.41 1 5 8 . 6 9 159.16 173.69 180.08 185.69 195.64 30 ° 143.88 145.99 161.34 166.29 168.04 182.53 186.03 190.33 45 ° 150.53 153.83 154.80 157.74 171.99 184.73 1 8 5 . 2 8 187.90


where

2 B .b 12 ( j0020 , ,-2000, Ry O o ~ w~wl -t- ~wiwj')

aZB22 (¢ooo2 ~_ ¢0200] R y O o ~= w~,ll -- ~ wiwj J

D12 (,ff0220 _ ¢-2002, I . bZDll j2o20 +_~oo~ wiw/ -~- -t- a2 Do wzwj --w~wj /

2 b D l 6 ( ¢-11.2Q ¢-2011 ] a2D22 j0202 ~ ~ - o \--wtwj -~- ~winj l -~- b2Do w~wj

2aD26 (¢1102 ¢0211] 4D66 ¢11.11 -]- ~ \~wiwj + ~wtwj ) ~- Do ~w,~j

uu = ¢ 0000. vv = .¢ 0000 mij ~uiuj , mij vvivj

ww ¢- 0000 mij = ~wi . j

2abBl6 (jOOll ¢11.o0~ Rx Do ~ w~wj + --w~,,~ /

2abB26 ( ¢OOll ¢11oo,~ Ry D O ~'vwiwj ~- ~wlWJ /

(20f)

(21a, b)

(21c)

J,,id~g= J.I od+eo~(~'~) of+gq3~(£'q) d~d~ 7 (22a) A O~ d 011 e O~ f O~ g

~defg J I od+e~(~'~l) of+g(gf(~'~) d~d~ (22b) uivj = A O~ d 017 e O~ f 01~ g

0d+e0y(~, ~) o f +g ffgj' ( ~, ~) f f j defg. = J JA d~Cdr/ (22c) u,wj o~d o~]e O~ f 07]g

jdefg I I od+eoff(~'lT) of+gff);:(~'~)d~dl] (22d) vivj = A O~ d O~] e O~ f 07] g

viwj "= A O~ d 0?7 e O~ f Oil g

= I I Oa+e~;'(~'~7) of+gqsf'(~'71) d~d~7 (22f) j aefg. . . . . j A O~ d 0?7 e O~ f 097g

and i, j = 1 , 2 , . . . , m, where m depends on the degree of polynomial . The reference plate flexural rigidity is D o = Q11h3/12 = El lh3/12(1 - ul2u2a). The non-dimensional frequency parameter is

A = wab p/ph (23) V'-'o

4 V E R I F I C A T I O N OF A N A L Y S I S

Two c o m m o n types o f composite materials have been

162


Table 9 Frequency parameters A ~ ~ab PX/ -~o for SSSS singly curved and doubly curved, 4-ply laminated graphite/epoxy shells with a/h = 100.0 and a/b = 1.0

Mode sequence number b R~.

SS Ry Rx ~ 1 2 "3 4 5 6 7 8

(-f l , fl. fl, fi) 0.0 all 0 ° 11.290 17.132 28.692 40.740 45.159 45.784 54.083 68.149 15 ° 11.675 19.276 32.209 39.432 46.737 50.011 59.851 71.181 30 ° 12.533 23.794 36.477 39.987 51.369 60.388 73.635 74.931 45 ° 12.986 26.562 34.427 44.379 57.181 65.775 66.299 83.097

0.5 -1.0 0 ° 61.698 73.436 92.271 93.196 93.619 102.99 114.36 118.11 15 ° 65.943 81.251 91.290 92.457 106.23 108.19 128.76 130.26 30 ° 69.606 78.857 89.271 91.283 108.52 108.79 110.91 115.43 45 ° 71.607 73.480 78.470 79.512 90.753 90.847 95.035 106.12

0.0 0 ° 30.220 31.125 44.231 48.040 56.491 56.818 61.774 69.385 15 ° 36.416 36.781 47.988 54.319 59.060 60.888 68.203 74.677 30 ° 48.792 49.089 63.786 65.675 70.724 72.363 84.200 91.923 45 ° 61.240 61.590 85.291 89.037 89.738 90.421 101.10 104.76 60 ° 64.186 71.764 93.764 98.357 119.57 120.56 122.19 122.62

.75 ° 56.843 79.494 89.134 92.137 111.41 113.96 135.65 137.11 90 ° 50.807 73.522 84.887 92.198 97.049 104.88 119.27 122.44

1.0 0 ° 68.082 87.778 96.115 103.55 108.25 115.52 127.94 131.61 15 ° 76.857 94.626 106.49 108.15 126.84 129.27 148.39 149.32 30 ° 95.536 95.842 121.52 128.61 133.88 135.31 143.50 147.95 45 ° 105.16 109.21 113.90 114.92 134.16 135.79 139.31 142.11

( - f i , ~ , - f l , fl) 0.0 all 15 ° 11.877 19.924 33.384 38.662 46.997 52.263 60.445 76.556 30 ° 13.180 25.890 35.598 44.839 51.633 70.527 71.984 74.039 45 ° 13.843 31.502 31.502 54.052 58.359 58.975 84.942 84.942

0.5 -1 .0 15 ° 65.810 84.450 88.718 94.280 107.42 109.91 129.99 131.85 30 ° 70.464 82.692 89.934 94.139 110.62 111.33 112.07 116.75 45 ° 75.536 75.903 75.903 84.051 90.438 92.020 105.59 105.59

0.0 15 ° 36.414 37.047 48.201 55.217 58.948 61.207 68.978 78.066 30 ° 49.410 49.671 64.395 65.520 74.027 75.016 86.699 90.657 45 ° 61.897 62.045 85.524 90,333 92.127 99.732 99.985 102.50 60 ° 64.201 71.636 98.385 99.630 119.82 122.38 125.87 126.84 75 ° 56.431 79.592 87.785 93.686 114.28 114.56 137.52 138.66

1.0 15 ° 76.358 93.404 106.14 108,80 129.15 129.91 149.95 150.13 30 ° 94.593 95.476 124.90 130,59 133.30 136.11 138.26 151.80 45 ° 105.06 108.67 113.26 115.29 131.98 140.37 142.17 145.84


chosen for this study: E-glass/epoxy (E/E) and graphite/ epoxy (G/E). The material properties of these composites are:

E-glass/epoxy (E/E)

graphite/epoxy (G/E)

E11 = 60.7GPa

E22 = 24.8 GPa

G12 = 12.0GPa

ul2 -- 0.23

Ell = 138.0GPa

E22 ~ 8.96 GPa

GI2 = 7.1 GPa

u12 ---- 0.30

4.1 Convergence o f eigenvalues

The global pb-2 shape function (13) is an infinite series. The orthogonal terms are generated according to the recurrence procedure outlined in equation (14a-c). In computation, it is only possible to work on a truncated series and the number of terms necessary for solution can be determined by performing convergence

tests of eigenvalues. For descriptive purposes, the first six terms of the pb-2 shape functions are expressed as follows:

(91(x,Y) = (gb(x,y) (24a)

(92(x, y) = x(91 (x, y) - ,~21 (91 (x, y) (24b)

(93(x,y) = y(91(x,y) - E31(91(x, y) - =32(92(x,y) (24c)

(94 (X, y ) = X 2 (91 (X, y ) -- ~'41 (91 (X, y ) -- ~'42 (92 (X, y )

- E43(93 (x, y) (24d)

(95 (X, y) = xy(91 (X, y ) -- ~'51 (91 (X, y ) -- "~52(92 (X, y )

-- ~53(93 (X~ y ) -- ~54(94(X, y ) (24e)

(96 (X~ y) = y 2(91 (x , y) - Z61 (91 (x , y) - Z62(92 (X, y )

-- ~'63 (93 (X, y ) -- ~64 (94 (X, y ) -- ~65 (95 (X, y ) .

(24f)

The in-plane and transverse basic functions (15a-c) are general because no restriction is placed on the geometric configurations and boundary constraints of the shells. In this study, two classes of boundary conditions are investigated: the simply supported (SSSS) and fully clamped (CCCC) shallow shells for both E/E and G/E.

163


Table 10 Frequency parameters ~ = co ab ~ for SSSS singly curved and doubly curved, 8-ply laminated graphite/epoxy shells with a/h = 100.0 and a/b = 1.0

Mode sequence number b Ry

SS Ry R x /3 1 2 3 4 5 6 7 8

(-/3,/3,-/3,/3,/3, -/3,/3, - /3) 0.0 all

0.5 -1 .0

(-/3,3,-3,/3,-/3,/3,-/3,/3) 0.0 all

0.0

1.0

0.5 -1 .0

0.0

1.0

0 ° 11.290 17.132 28.692 40.740 45.159 45.784 54.083 68.149 15 ° 12.032 20.205 33.746 39.896 48.134 52.579 61.911 76.215 30 ° 13.431 26.414 37.464 45.452 54.068 70.427 76.285 78.011 45 ° 14.089 30.901 34.633 53.446 61.329 64.467 81.810 91.606

0 ° 61.698 73.436 92.271 93.196 93.619 102.99 114.36 118.11 15 ° 66.637 84.775 91.540 95.504 109.04 111.17 131.79 133.29 30 ° 71.389 84.010 92.242 95.611 111.80 112.40 112.97 119.33 45 ° 75.055 75.602 78.056 84.294 93.836 93.879 102.82 112.28

0 ° 30.220 31.125 44.231 48.040 56.491 56.818 61.774 69.385 15 ° 36.982 37.628 48.331 56.010 59.528 62.241 70.365 78.293 30 ° 50.454 50.933 64.789 66.677 75.657 76.380 89.479 94.027 45 ° 63.235 64.373 86.669 91.650 93.478 100.38 101.63 108.95 60 ° 65.515 74.781 100.70 101.44 121.04 124.28 127.78 128.59 75 ° 57.347 81.746 89.142 95.493 115.53 116.01 139.04 139.99 90 ° 50.807 73.522 84.887 92.198 97.049 104.88 119.27 122.44

0 ° 68.082 87.778 96.115 103.55 108.25 115.52 127.94 131.61 15 ° 77.187 95.871 107.08 110.79 130.61 130.92 151.19 151.20 30 ° 96.742 97.386 127.01 132.33 134.74 137.66 145.89 152.28 45 ° 107.01 110.47 116.12 116.48 138.64 139.08 144.30 147.98 15 ° 12.078 20.354 34.017 39.701 48.186 53.086 62.057 77.582 30 ° 13.570 26.885 37.208 46.600 54.007 73.209 75.478 77.551 45 ° 14.272 32.952 32.952 56.753 61.493 61.646 89.466 89,466 15 ° 66.593 85.845 90.277 96.321 108.64 112.15 131.53 134,10 30 ° 71.520 85.008 91.627 96.563 111.68 112.59 114.73 118,68 45 ° 75.839 76.691 76.691 86.048 92.188 94.021 109.30 109,30 15 ° 36.977 37.685 48.381 56.211 59.491 62.307 70.547 79.063 30 ° 50.501 51.100 64.925 66.615 76.015 77.331 90.034 93.431 45 ° 63.261 64.499 86.647 91.745 94.043 101.12 104.77 106.21 60 ° 65.483 74.700 101.58 101.72 121.00 125.55 128.31 128.97 75 ° 57.241 81.830 88.712 95.886 115.48 116.82 138.97 140.74 15 ° 77.062 95.566 106.97 110.95 130.99 131.19 151.25 151.57 30 ° 96.636 97.127 127.76 132.78 134.40 137.98 144.39 153.75 45 ° 106.96 110.31 115.80 116.64 137.02 141.16 144.65 150.12


The basic functions for the SSSS shell are

~b~(~, 7) = ~ff(~, 7) = ~ ' (~ , 7) = (~2 _ 0.25)(72 _ 0.25)

(25)

and for the CCCC shell are

q~/~(~, 7 ) = (~2 _ 0 . 2 5 ) ( 7 2 _ 0.25) (26a)

4~ff(~, 7) = (~2 _ 0.25)(72 _ 0.25) (26b)

~ ' (~ , 7) = (~2 _ 0.25)2(72 _ 0.25)2. (26c)

Following the procedure described above, a convergence study of eigenvalues has been carried out and shown in Table 1 (SSSS) and Table 2 (CCCC) for a representative 8-ply E/E shell with symmetric and unsymmetric laminate stacking sequences. The first eight frequencies are presented in an ascending order. The power of polynomial p and hence the number of terms m, which are related by equation (17), for q~, 0~ and 0F (i = 1 , . . . , m) are mutually independent. These terms are increased steadily to determine the number of terms required to achieve numerical convergence.

It is widely known that the Ritz method overestimates the structural stiffness and frequency. A higher number of terms in the shape functions tends to decrease the eigenvalues as observed in Tables 1 and 2. Although

absolute convergence can never be achieved, it is obvious tha tp = 10 or m = 66 for u, v and w is adequate to obtain convergent eigenvalues to five significant figures. It is equivalent to solving a matrix with a determinant size (66 × 3) × (66 x 3). Unless otherwise stated, a total of 66 terms for u, v and w are used in the subsequent computation.

4.2 Comparison study

Having determined the number of terms required to obtain convergent eigenvalues, a comparison study has been carried out to verify the computational results. Comparisons include frequencies for laminated flat plates, cylindrical and doubly curved shells. The frequencies are expressed in cycles per second (Hz).

Tables 3 and 4 compare laminated plate results with Jensen et al. 6, Crawley 13 and Qatu and Leissa 16. In Table 3, the comparison for cantilevered plates having six plies with symmetric lamination is presented. The analytical results of Jensen et al. 6 were obtained using the finite element method (FEM) and the Ritz method with different combinations of trial functions (A, B, C, D, A', and D') . The results of Qatu and Leissa 16 were obtained using the Ritz method with simple two- dimensional polynomials as the displacement functions

164

Vibration of composite shallow shells." K. M. Liew et al.

Table 11 Frequency parameters ~ = wabx/~/D o for CCCC singly curved and doubly curved, 4-ply laminated E-glass/epoxy shells with a/h = 100.0 and a/b = 1.0


SS R>. Rx ~ 1 2 3 4 5 6 7 8

(-~,fl .H,-fi) 0.0 all 0 ° 29.104 50.829 67.287 85.681 87.143 118.57 126.14 136.86 15 ° 28.957 51.565 65.973 85.103 89.573 119.92 122.80 139.43 30 ° 28.667 53.633 62.743 85.102 95.140 114.27 122.83 139.03 45 ° 28.523 55.569 60.226 85.248 102.57 105.22 124.55 137.57

0.5 -1.0 0 ° 118.39 124.78 136.27 138.25 158.83 165.61 185.93 187.01 15 ° 118.41 125.29 135.29 137.29 160.28 164.68 180.80 181.40 30 ° 118.53 124.32 131.32 134.35 160.59 160.71 167.51 172.75 45 ° 119.32 122.86 127.22 132.17 157.00 159.22 160.66 169.39

0.0 0 ° 70.128 80.489 112.33 112.76 114.66 135.67 139.81 161.89 15 ° 72.357 82.172 113.80 114.87 115.81 136.91 143.20 160.32 30 ° 77.278 86.891 118.37 119.00 121.11 141.34 152.94 157.35 45 ° 82.169 93.953 123.80 124.99 133.38 149.07 158.03 167.09 60 ° 84.912 101.90 126.47 131.09 150.21 160.25 163.90 169.69 75 ° 85.106 108.51 124.73 133.70 164.54 164.65 167.52 174.65 90 ° 84.666 111.32 122.57 134.26 162.24 166.95 170.82 179.45

1.0 0 ° 138.11 155.33 159.93 163.73 182.53 186.66 199.10 206.03 15 ° 140.53 154.86 164.51 164.54 181.16 190.04 198.01 205.34 30 ° 146.82 155.07 163.15 171.85 174.78 192.28 198.80 203.49 45 ° 154.45 156.31 156.80 168.93 175.34 192.78 199.42 202.18

(-fl, f l , -~,~) 0.0 all 15 ° 28.907 51.652 65.673 85.985 88.804 120.75 122.17 139.33 30 ° 28.571 54.054 61.951 86.715 93.986 112.95 126.15 137.50 45 ° 28.425 57.718 57.718 87.127 102.29 102.82 132.31 132.31

0.5 -1.0 15 ° 118.72 126.26 135:15 136.29 160.46 164.77 180.81 181.49 30 ° 119.75 127.60 130.52 131.45 160.64 162.75 167.24 171.89 45 ° 123.51 123.51 127.71 128.09 157.09 160.10 165.22 165.22

0.0 15 ° 72.348 82.192 114.09 114.56 115.79 137.67 142.56 159.90 30 ° 77.271 86.970 119.00 119.03 120.53 143.57 151.28 156.62 45 ° 82.129 94.048 124.13 125.23 133.07 152.10 157.50 166.04 60 ° 84.777 101.94 126.38 131.53 150.00 163.24 163.67 168.08 75 ° 84.988 108.55 124.50 133.93 164.55 164.76 167.52 174.71

1.0 15 ° 140.42 154.80 164.35 164.58 181.09 190.49 197.83 205.37 30 ° 146.73 155.02 163.13 171.79 174.58 195.23 197.44 203.47 45 ° 155.16 156.00 156.43 168.74 175.25 196.86 198.14 202.18

ss: Stacking sequence of laminates

and a total o f 64 terms (only for w). Thei r analysis did

not account for in-plane displacements in this case. It can

be seen that excellent agreement has been achieved

a m o n g the compu ta t i ona l solutions. The congruence o f

the results o f Qa tu and Leissa 16 and the pb-2 solut ions is

par t icular ly noted. The exper imenta l frequencies o f

Jensen et al. 6 are also in very good agreement with the

pb-2 solutions.

In Table 4, the F E M and exper imenta l results o f Crawley 13 and those o f Qa tu and Leissa 16 have been used

for compar i son . The plates are composed o f eight-ply

laminates with symmetr ic laminat ion. Two different

plate geometr ies with aspect rat ios a/b = 1.0 and 2.0

have been studied. Again , it is observed that excellent

agreement has been achieved for the F E M , Qatu and

Leissa 16 and the pb-2 solutions. Agreemen t with the

exper imenta l frequencies is general ly satisfactory, allow-

ing for uncertaint ies associated with experiments .

The compar i son for l amina ted shell frequencies is

shown in Tables 5 and 6 for singly curved and doubly

curved shal low shells, respectively. Table 5 shows the results o f var ious me thods for cant i levered cylindrical

shells having eight plies with symmetr ic laminat ion. The F E M and exper imenta l results are due to Crawley 13.

G o o d agreement a m o n g the results can be observed. It is interest ing to note that the solut ions o f Qa tu and Leissa 16

for the [45 °, - 4 5 °, - 4 5 °, 45°]sym laminated shell include

an addi t ional mode-5 f requency which is not found in

o ther analyses.

Table 6 presents the compar i son for doubly curved

graphi te /epoxy shells with six plies in symmetric , angle-

ply laminat ion. The shells are complete ly free. Shells with

both posi t ive and negat ive Gauss ian curvatures (Rx/Ry) are investigated. In this case, only the results o f Qa tu and

Leissa 17 are available. The first 10 frequencies are shown

in an ascending order. The shell with Rx/Ry = 0 represents a cylindrical shell where Rx << Ry ~ oo. The

shell geomet ry ranges f rom low to high shallowness rat io

(a/Rx). The frequency pa ramete r is expressed as

cvaZv/p/Ell h2. As a whole, this compar i son presents

excellent agreement between the results.

5 P A R A M E T R I C S T U D Y

The numerica l study covers the E-glass /epoxy and

graphi te /epoxy compos i te shells with 4-ply symmetric; 4-ply unsymmetr ic ; 8-ply symmetric; and 8-ply unsym-

metr ic laminat ions . The first eight frequencies are

tabula ted in ascending order for two boundary classes

in the fol lowing tables. The non-d imens iona l f requency

165

Vibration of composite shallow shells." K. M. Liew et al.

Table 12 Frequency parameters A = a ; a b p x / ~ o for CCCC singly curved and doubly curved, 8-ply laminated E-glass/epoxy shells with a/h = 100.0 and a/b = 1.0

b R3 Mode sequence number

SS Ry R x 3 1 2 3 4 5 6 7 8

( -3 ,3 , -3 ,3 ,3 , -3 ,3 , -3 )

( -3 ,3 , -3 ,3 , -3 ,3 , -3 ,3 )

0.0 all 0 ° 29.104 50.829 67,287 85.681 87.143 118.57 126.14 136.86 15 ° 29.018 51.740 66,040 86.026 89.178 120.90 122.90 139.53 30 ° 28.841 54.309 62.776 87,067 94.847 114.44 126.28 139.38 45 ° 28.751 57.213 59,515 87.628 103.37 104.47 130.18 136.85

0.5 -1 .0 0 ° 118.39 124.78 136.27 138.25 158.83 165.61 185.93 187.0l 15 ° 118.86 126.28 135.39 136.74 160.69 164.96 181.09 181.84 30 ° 119.87 127.13 131.11 132.79 161.09 163.00 167.92 173.14 45 ° 121,95 125.79 126.05 130.60 157.74 160.84 163.54 168.76

0.0 0 ° 70,128 80,489 112.33 112.76 114.66 135.67 139.81 161.89 15 ° 72,438 82.287 114.21 114.79 116.03 137.75 142.89 160.47 30 ° 77,545 87.284 119.43 119.50 120.98 143.76 152.48 157.67 45 ° 82,583 94.641 124.80 125.69 133.60 152.37 158.32 168.18 60 ° 85,280 102.66 127.00 131.99 150.53 163.15 164.12 169.50 75 ° 85,247 108.91 124.78 134.17 164.80 165.00 167.72 174.97 90 ° 84.666 111.32 122.57 134.26 162.24 166.95 170.82 179.45

1.0 0 ° 138.11 155.33 159.93 163.73 182.53 186.66 199.10 206.03 15 ° 140.63 155.21 164.47 164.68 181.23 190.73 198.18 205.59 30 ° 147.08 155.59 163.30 172.25 174.97 195.32 198.65 203.86 45 ° 155.27 156.43 156.80 169.41 175.81 196.44 199.92 202.59

0.0 all 15 ° 29.005 51.762 65.965 86.298 88.934 121.11 122.75 139.49 30 ° 28.816 54.415 62.574 87.524 94.498 114.11 127.20 138.88 45 ° 28.725 58.316 58.316 88.132 103.30 103.83 133.82 133.82

0.5 -1 .0 15 ° 118.93 126.54 135.35 136.48 160.74 164.99 181.09 181.86 30 ° 120.17 128.32 130.90 131.69 161.10 163.56 167.84 172.86 45 ° 123.92 123.92 128.40 128.44 157.76 160.69 166.37 166.37

0,0 15 ° 72.436 82.292 114.34 114.65 116.02 137.97 142.70 160.37 30 ° 77.542 87.302 119.50 119.69 120.74 144.43 151.96 157.49 45 ° 82,570 94.661 124.88 125.74 133.51 153.24 158.19 167.53 60 ° 85.245 102.67 126.98 132.09 150.47 164.06 164.13 168.86 75 ° 85.217 108.92 124.73 134.22 164.77 165.06 167.72 174.99

1,0 15 ° 140.60 155.20 164.43 164.69 181.21 190.85 198.13 205.59 30 ° 147.06 155.58 163.30 172.23 174.92 196.65 197.73 203.85 45 ° 155.63 156.04 156.83 169.36 175.78 198.23 198.87 202.58


parameter is expressed as A = a;abx/~/D o, where a; is the physical frequency of vibration in radians per second. Assuming constant shell span a, width b, density p, thickness h and reference flexural rigidity Do, A is directly proport ional to a~. Therefore, all the conclusions drawn in the following sections for A also apply to co.

5.1 Simply supported shells (SSSS)

Tables 7 and 8 show the non-dimensional vibration frequency parameters for E/E shells with 4-ply and 8-ply, respectively. The corresponding results for G/E shells are presented in Tables 9 and 10. For each of the tables, the angle of lamination (/3) ranges from 0 ° to 90 ° in 15 ° increments; the curvature ratio (Ry/Rx) varies f rom - 1.0 to 1.0 and the shell shallowness ratio (b/Ry) changes from 0.0 (a flat plate for non-infinite Ry/Rx) to 0.5 (doubly curved shallow shell). For a cylindrical shallow shell (singly curved), b/Ry = 0.5 and Ry/Rx = 0.0, where the radius of curvature R x is infinite.

For all cases except the cylindrical shells, the values of A for/3 = 0 ° and 90°; 15 ° and 75°; and 30 ° and 60°; are exactly identical because these pairs of/3 refer to shells with equivalent laminations due to symmetries and shell geometry (a/b = 1.0 and Ry/R x = - 1 . 0 or 1.0) and

boundary conditions (SSSS). For shells with 4-ply and 8-ply laminations (Tables 7-10), it is irrelevant whether the stacking sequences are symmetric or unsymmetric for /3 = 0 ° and 90 ° because the fibres are parallel to each other. As a result, the repeated solutions are not listed in these tables.

F rom Tables 7 and 8, increases in/3 cause fluctuations in A. There is no general trend whether A should increase or decrease. However, it is interesting to note that the fundamental A increases for increasing/3 from 0 ° to 45 ° and decreases thereafter from 45 ° to 90 ° , are restricted to shells with symmetry in geometry and boundary conditions (i.e. excluding the cylindrical shells). The maximum A occurs when /3 - -45 °. For the cylindrical shells (b/Ry = 0.5 and Ry/Rx = 0.0), the highest A is shifted to approximately/5 = 60 °. It can be concluded that the lack of geometry symmetry for E/E shells (SSSS) causes unsymmetric distribution of fundamental vibration frequencies with respect to/3.

The above conclusion can also be extended to G/E shells (SSSS) as observed in Tables 9 and 10. The maximum A also occurs when/3 = 45 °. The symmetry in distribution of fundamental A with respect to /3 breaks down for cylindrical shells.

The significance of number of plies in symmetric and unsymmetric laminations can be examined in Tables 7

166


Table 13 Frequency parameters A = ~abx/~/D o for CCCC singly curved and doubly curved, 4-ply laminated graphite/epoxy shells with a/h = 100.0 and a/b = 1.0


SS Ry Rx /3 1 2 3 4 5 6 7 8

( - f l , 3 , / 3 , - / 3 ) 0.0 all 0 ° 23.853 29.720 41.728 59.971 62.976 67.470 76.464 83.992 15 ° 23.288 30.993 44.975 60.089 64.875 67.208 80.104 89.074 30 ° 22.226 35.095 53.402 54.198 68.418 77.777 92.425 99.266 45 ° 21.740 39.207 48.080 60.123 75.333 84.410 85.309 104.10

0.5 -1 .0 0 ° 76.398 84.587 102.02 110.48 111.58 120.57 123.18 134.89 15 ° 77.424 89.273 103.02 107.27 117.95 118.84 139.39 140.19 30 ° 76.611 85.295 96.824 96.954 114.20 115.39 122.92 129.16 45 ° 74.592 75.414 83.962 86.226 102.43 103.91 108.30 121.88

0.0 0 ° 37.669 40.228 50.251 61.716 74.408 74.857 81.262 85.153 15 ° 42.450 44.227 53.959 67.733 74.342 77.112 86.118 91.355 30 ° 53.417 55.183 68.400 75.833 84.156 85.373 100.20 110.70 45 ° 65.547 70.454 89.313 93.350 97.140 104.18 112.04 122.34 60 ° 70.808 85.006 104.02 104.30 123.41 125.01 131.26 133.00 75 ° 69.940 91.445 102.82 106.54 122.42 124.60 145.71 146.51 90 ° 68.339 85.251 105.11 105.98 111.49 122.88 128.04 138.90

1.0 0 ° 82.033 97.804 114.49 116.35 120.56 131.32 136.25 145.98 15 ° 86.790 107.71 115.96 120.94 136.03 137.53 155.83 156.28 30 ° 99.082 107.99 129.20 131.81 136.73 142.35 150.23 157.00 45 ° 113.66 114.81 116.57 116.81 142.26 143.11 147.39 154.12

( - /3 ,3 , - /3 , /3 ) 0.0 all 15 ° 22.751 31.142 45.726 58.178 66.345 66.371 80.156 92.747 30 ° 21.863 36.198 51.182 57.858 67.434 86.681 91.308 94.702 45 ° 21.593 43.491 43.491 68.417 75.424 75.822 103.24 103.24

0.0 -1 .0 15 ° 76.499 91.632 103.45 108.70 113.40 122.88 137.69 142.58 30 ° 77.101 89.863 95.675 99.485 115.66 120.09 120.57 129.95 45 ° 75.806 80.221 80.221 91.694 102.21 102.95 119.61 119.61

0.0 15 ° 42.494 44.261 54.107 68.809 72.784 76.550 86.298 94.102 30 ° 53.909 55.727 69.217 74.764 84.893 90.013 101.16 109.44 45 ° 65.903 71.238 89.307 94.905 98.507 108.78 116.14 121.70 60 ° 70.274 85.572 104.07 107.47 123.53 130.52 131.92 133.17 75 ° 68.541 94.253 97.698 109.57 120.15 127.76 144.95 148.37

1.0 15 ° 85.510 108.89 112.21 122.50 134.91 139.61 156.54 156.57 30 ° 98.461 106.86 131.11 133.19 135.99 144.46 156.41 157.45 45 ° 113.33 114.26 116.59 116.84 144.42 149.88 150.02 155.44


and 8, and Tables 9 and 10. Because higher mode frequencies involve uncertainties such as the switching of mode sequences, the effect of number of plies has been studied based on the fundamental mode only. It is expected that shells with more plies tend to have higher stiffnesses and fundamental A. It is evident that A increases with increasing number of plies in Tables 7 and 8, and Tables 9 and 10. This effect is much more pronounced for the G/E shells in Tables 9 and 10 than the E/E shells in Tables 7 and 8.

The effect of shallowness ratio and shell curvature is obvious from Tables 7-10. Increases in shell curvature in the x- or y-directions result in higher fundamental A. This is expected because shells with higher curvature are stiffer. This effect has been fully studied by Lim and Liew 19 for isotropic singly curved shallow shells and Liew and Lim 2° for isotropic doubly curved shallow shells.

To enhance vibration visualization, some typical contour vibration mode shapes are illustrated in Figures 2 and 3 for the E/E 4-ply symmetric and unsymmetric shells. Only the transverse or W-mode shapes are presented because these vibration modes are dominant over the in-plane modes. The shaded portions represent regions with positive displacement amplitude. Positive and negative displacement amplitude regions can be

interchanged because only the relative amplitudes are of much concern. The lines of demarcation are the nodal lines with zero amplitude of vibration. The effects of various parameters can be easily seen in these mode shape figures. Mode shape symmetries with respect to the x- and y-axes are observed for /3 = 0 °. They are also observed for /3 = 90 ° because the fibres are parallel. These symmetries disappear for laminates with non- parallel angles of lamination (0 </3 < 90°).

5.2 Fully clamped shells (CCCC)

The vibration frequencies corresponding to the values in Tables 7-10 for the SSSS shells are shown in Tables 11-14 for the CCCC shells. Tables 11 and 12 are results for the E/E shells and Tables 13 and 14 for the G/E shells.

Unlike the SSSS shells, the CCCC shells reflect different effects of/3 on the fundamental A. Increases in /3 result in lower and higher fundamental A for various shell configurations. Generally, increases in/3 from 0 ° to 45 ° render higher fundamental A for cylindrical (b/Ry = 0.5 and Ry/R x = 0.0) and spherical shells (b/Ry = 0.5 and Ry/Rx = 1.0). For other shell geometries, the fundamental A fluctuates. The maximum fundamental A could be approximately 75 ° for the CCCC cylindrical

167


Table 14 Frequency parameters A = wabx/-~-/D o for CCCC singly curved and doubly curved, 8-ply laminated graphite/epoxy shells with a/h = 100.0 and a/b = 1.0


SS Ry Rx /3 1 2 3 4 5 6 7 8

( - / 3 , 3 , - / 3 , / 3 , / 3 , - / 3 , / 3 , - / 3 ) 0 .o all

0.5 -1 .0

( - / 3 , / 3 , - / 3 , / 3 , - / 3 , / 3 , - 7 , 3 ) 0 .0 all

0.0

1.0

0.5 -1 .0

0.0

1.0

0 ° 23.853 29.720 41.728 59.971 62.976 67.470 76.464 83.992 15 ° 23.598 31.904 46.499 60.626 67.136 68.655 82.478 93.247 30 ° 23.109 37.731 54.499 59.609 71.389 87.918 96.811 101.02 45 ° 22.887 43.954 48.127 69.745 79.652 82.987 101G9 112.45

0 ° 76.398 84.587 102.02 110.48 111.58 120.57 123.18 134.89 15 ° 78.262 92.592 106.68 108.83 117.07 123.96 140.25 143.91 30 ° 78.953 90.559 98.630 100.79 117.48 122.10 123.93 134.84 45 ° 76.086 79.673 83.508 93.134 106.01 107.23 118.13 128.45

0 ° 37.669 40.228 50.251 61.716 74.408 74.857 81.262 85.153 15 ° 43.139 44.879 54.726 69.651 74.782 78.567 88.486 94.786 30 ° 55.106 56.777 70.725 76.992 87.753 92.387 105.41 114.91 45 ° 67.634 73.414 90.663 96.733 101.10 112.62 117.35 129.60 60 ° 72.520 88.806 106.77 109.39 125.00 132.43 132.80 137.60 75 ° 70.563 94.877 101.83 111.18 122.60 129.16 147.12 149.61 90 ° 68.339 85.251 105.11 105.98 111.49 122.88 128.04 138.90

0 ° 82.033 97.804 114.49 116.35 120.56 131.32 136.25 145.98 15 ° 87.240 110.79 114.92 124.51 136.70 140.92 157.54 158.28 30 ° 100.23 110.45 132.61 135.03 137.54 147.25 155.92 161.52 45 ° 115.33 117.19 118.10 118.37 146.41 152.94 153.47 157.58 15 ° 23.463 31.929 46.662 60.145 67.457 68.437 82.480 94.051 30 ° 22.997 37.937 53.923 60.401 71.045 90.221 96.077 99.775 45 ° 22.814 45.928 45.928 72.443 79.537 79.956 109.28 109.28 15 ° 78.018 93.221 105.64 110.94 115.14 125.02 139.75 144.40 30 ° 78.979 91.545 98.292 101.14 117.30 123.21 124.33 134.52 45 ° 76.251 81.631 81.631 94.770 105.30 106.00 124.84 124.84 15 ° 43.141 44.874 54.756 69.892 74.381 78.414 88.515 95.378 30 ° 55.182 56.853 70.881 76.688 87.850 93.382 105.53 113.86 45 ° 67.656 73.525 90.596 96.982 101.46 111.63 121.52 127.77 60 ° 72.343 88.895 106.66 109.89 124.96 132.46 135.80 136.45 75 ° 70.212 95.855 100.26 112.09 121.79 129.94 146.91 149.97 15 ° 86.920 111.75 113.30 124.96 136.29 141.41 157.66 158.41 30 ° 100.05 110.15 132.90 135.44 137.12 147.64 159.38 160.06 45 ° 115.23 117.00 118.14 118.27 146.71 154.51 154.56 158.87


W - M o d e Shapes

M o d e 1 M o d e 2 M o d e 3 M o d e 4 M o d e 5 M o d e 6 M o d e 7 M o d e 8

OO

l i i ~ i : : F E : : i i i ~ i i ~ l ~ i ' I CS 3 I ~ i ~ 1 l ~ ~ ~ l l

.... I ~ : I ~ C . ~ i i ~ ::ii~:~:::::::i::::~ii:: i i ~ ~

i ~ _ i t ~ i ! i i ~ i l i ~ i :7::~i ! ~ i i ~ ~ i ~ i i , :i!iii::7:i:i

131.82- ~ilii-]42-83- ,- -1-5 3. (0 , --1]ff.7-2-~ -7/-4]7- -175_78- --]8-7_70-~ --196~14- ~:::~=~:~:~ 7 ::~ . . . . . . ~ : ~ -~ - - ~ - "= :~ i - ~ - ~ - : ~ . . . . . . t . . . . . . " - - ~ = ~ : - -~::~:~: ii! - -

--~-5_~- --~_~- --I~,3-~- --17~,6-~- -7/-377>7- -]7-~7-7~- -q~5_~- --i~-~- , . . . . . . ~ ~ ~ : ~ , - - - ~ < . , ~ , - - ~ ~ , . . . . . . . . . ~ -~ :~ :~ , ,-~z- ~-_-:~

. . . . . .

--i~-3.~7- --i75T~- --xi£6~- -q~ff.6-3- -7(87i~- -Tffo_og- -qi5To~- - q i o ~ 6 - 0 -

I : ....... ~ I: I I rl i - - I I t !

15 °

30 °

45 °

150.20 153.05 155.25 157.39 171.49 181.19 183.25 187.61

Figure 2 Contour W-mode shapes for vibration of SSSS 4-ply symmetrically laminated E-glass/epoxy shells with a/h = 100.0, a/b -- 1.0, b/Ry = 0.5, Ry/Rx = 1.0 and stacking sequence ( /3,/3,/3,-/3)

168


W-Mode S h a p e s

Mode 5 Mode 6 Mode 7 Mode 8

1 5 °

3 0 °

4 5 °

M o d e 1

.........

QI:::I 143.58

150.12

Mode 2 Mode 3 _ Mode 4

II I diiiii?

ii I 143.07 158.54

...... .... 145.45 161.15 166.02

153.08 155.00 157.30

I

I 167.77

i

181.19

I I

I

195.16

189.24

Ii 1 i

171.58 183.51 184.72 186.08

Figure 3 Contour W-mode shapes for vibration of SSSS 4-ply unsymmetrically laminated E-glass/epoxy shells with a/h = 100.0, a/b = 1.0, b/R~. - 0.5, R J R x = 1.0 and stacking sequence (-/3, /3,- /3, /3)

shells. The symmetry of fundamental frequency distribution with respect to/3 sill holds for shells with symmetric geometry (i.e. flat plate, hyperbolic paraboloidal and spherical shells).

The CCCC shells behave similarly to the SSSS shells when the number of plies is increased. Higher fundamental frequencies result in much more pro-

nounced effects for the G/E shells in Tables 13 and 14 than the E/E shells in Tables 11 and 12. The shallowness ratio and shell curvature also have similar effects on the fundamental A for both SSSS and CCCC shells. Increases in these factors result in shells with higher stiffness and thus higher vibration frequencies.

The contour W-mode shapes for the CCCC shells are

/~ W-Mode Shapes

Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6 Mode 7 Mode 8

,

3 0 c

4 5 c

138.11

140.53

146.82

//H//HZH/H// 154.45

155.33 159.93 163.73

154.86 164.51 164.54

155.07 163.15 / H H H H H J / / ~ /

~ H/IH////////H 156.31 156.80

171.85 "////H/x////.'//

168.93

/ H / / I H H H / / / /

182.53

, ' / / / H / / / I H / / / Z

181.16

174.78

175.34

/ / / / / / / ~ / ~ 1 / i / / /

M I H I I I I I H I I H

186.66

//H////z//////. 190.04

192.28

/ / / / / ~ / / t / / / / / / / / / / / / / / / / / / / / / / /

199.10

"/////HI/I/I//, 198.01

206.03

////I/I/z/I/H// 205.34

~ / ///// 198.80 203.49

192.78 199.42 202.18

F i g u r e 4 Contour W-mode shapes for vibration of CCCC 4-ply symmetrically laminated E-glass/epoxy shells with a/h = 100.0, a/b = 1.0, b/R~ = 0.5, R3/Rx = 1.0 and stacking sequence ( -~ , /3 , /3 , - /3 )

169

Vibration of composite shallow shells." K. M. Liew e t al.

W - M o d e S h a p e s

Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6 Mode 7 Mode 8

i : i~ i~ : i : i :~:::~+i~::i~ ~i :::: ~ i ~ ii ~ii:::~::~i~ ~ : ~ i ~ " ........... ::~:: ~ : : : : : ::: ::::::i~:: ...... ~i~ ~ @ ~ _ ~ :: 1 ~ ~ . . . . . . . . . :: i::~ i~::'~ii~t~::::::: ~i

, ~ . . . . . . . . . . . . . C) :; . . . . . . . . . . . . . . . . . . . . . . . . . .; ~ : . ~ : ~ 1 ~ ~ ~ - i ~ ~ , ~ 1 ~ ! ~ ~ E : : ~ t \ ~ i ~ ! . . . . , . : , : . £ ; ~ : ........................................ ~.:::.:..~ , ~ . . . ~ : - ~ + : : : : )) . . . . ........... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - > ) . . . . . . . . . . . . . . . . . . . .

140.42 154.80 164.35 164.58 181.09 190.49 197.83 205.37 / e L I I I H ( / ( /

[ ; ; ~ : ~ : : : : : : : : : ....... , : ~ : , / ; ~ I : ~ ' ' ~ , '~ : ::~ i ;

146.73 155.02 163,13 171.79 174.58 195.23 197.44 203.47 / H H / H / H / H / / / H H / H / / H / / / / " / H / / / / / / / / / / H / / / / H / M / H / H / ~ / / / / / / / / / / 1 / / / / H M H / / H ~ H H / H / / / / H H H H / / / / / / / / H / / / / / /

.............................................................. : ................................................................................................................... ................... .,,,,,.:...:...::.!! : . , . . . . . . ;

155.16 156.00 156.43 168.74 175.25 196.86 198.14 202.18

Figure 5 Contour W-mode shapes for vibration of CCCC 4-ply unsymmetrically laminated E-glass/epoxy shells with a/h = 100.0, a/b = 1.0, b/Ry = 0.5, R>./R x = 1.0 and stacking sequence (-fl , 3,-/3,/3)

illustrated in Figures 4 and 5 for 4-ply symmetric and unsymmetric laminations, respectively. The effects described above can be easily studied in these mode shapes. It is again observed that /3 ¢ 0 (and ¢ 90 °) results in unsymmetric mode shapes.

6 CONCLUSIONS

This paper presents a general method of solution for the analysis of vibration of thin, symmetric and unsymmetric laminated shallow shells with arbitrary boundary constraints. It extends and generalizes the previous analyses on vibration of doubly curved composite shallow shells, which were restricted to shells with simple boundary conditions and symmetric laminations. The extremum energy principle has been used to obtain the strain and kinetic energy integral equations. The kinematically oriented global shape functions developed allow a variety of boundary constraints monitored by appropriate basic powers.

Excellent convergence of eigenvalues has been shown with very low degrees of polynomial. The consistency and accuracy of the solutions using this approach have been verified through a series of comparison studies involving laminated plates and shells. New results for generally laminated shells (symmetric and unsymmetric) with multiple plies have been presented here for two types of composite materials. The results suggest that the lack of geometric symmetry causes unsymmetric distribution of fundamental vibration frequencies.

Typical transverse vibration mode shapes have been presented to enhance the numerical results. These illu- strations better reflect the effects of various geometry and lamination parameters of the shells. It has been shown

that symmetry in variation mode shapes breaks down for non-parallel angles of lamination.

REFERENCES

1 Bert, C.W. Dynamics of composite and sandwich panels. Part I. The Shock and Vibration Digest 1976, 8(10), 37-48

2 Bert, C.W. Dynamics of composite and sandwich panels. Part II. The Shock and Vibration Digest 1976, 8(11), 15-24

3 Bert, C.W. Research on dynamic behaviour of composite and sandwich plates V: Part I. The Shock and Vibration Digest 1991, 23(6), 3 14

4 Bert, C.W. Research on dynamic behaviour of composite and sandwich plates V: Part II. The Shock and Vibration Digest 1991, 23(7), 9-21

5 Baharlou, B. and Leissa, A.W. Vibration and buckling of generally laminated composite plates with arbitrary edge conditions. Int. J. Mech. Sci. 1987, 29, 545-555

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7 Liew, K.M. and Lim, C.W. Vibratory characteristics of general laminates--l: Symmetric trapezoids. J. Sound Vibration 1995, 183(4), 615-642

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9 Whitney, J.M. The effect of boundary conditions on the response of laminated composites. J. Compos. Mater. 1970, 4, 192-203

10 Whitney, J.M. 'Structural Analysis of Laminated Anisotropic Plates'. Technomic Publishing Company, U.S., 1987

11 Mirza, S. Recent research in vibration of layered shells. Trans. ASME. J. Pressure Vessel Tech. 1991, 113, 321 325

12 Kapania, R.K. A review on the analysis of laminated shells. Trans. ASME J. Pressure Vessel Techn. 1989, 111, 88 96

13 Crawley, E.F. The natural modes and frequencies of graphite/ epoxy cantilevered plates and shells. J. Compos. Mater. 1979, 13, 195 205

14 Qatu, M.S. Free vibration and static analysis of laminated composite shallow shells. Ph.D. Thesis, The Ohio State University, U.S.

15 Leissa, A.W. and Qatu, M.S. Equations of elastic deformation of laminated composite shallow shells. Trans. ASME J. Appl. Mech. 1991, 58, 181 188

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16 Qatu, M.S. and Leissa, A.W. Natural frequencies for cantilevered doubly-curved laminated composite shallow shells. Compos. Struet. 1991, 17, 227-255

17 Qatu, M.S. and Leissa, A.W. Free vibrations of completely free doubly curved laminated composite shallow shells. J. Sound Vibration 1991, 151(1), 9 29

18 Vinson, J.R. and Sierakowski, R.L. 'The Behaviour of Struc-

tures Composed of Composite Materials'. Martinus Nijhoff, Dordrecht, The Netherlands, 1986

19 Lira, C.W. and Liew, K.M. A pb-2 Ritz formulation for flexural vibration of shallow cylindrical shells of rectangular planform. J. Sound Vibration 1994, 173(3), 343-375

20 Liew, K.M. and Lim, C.W. Flexural vibration of doubly curved shallow shells. Acta Mech., in press

171

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Effects of general laminations and boundary constraints on vibration of composite shallow shells