Journal ofSound and Vibration (1987) 119(l),
INFLUENCE FREE
111137
OF THICKNESS VIBRATIONS
CYLINDRICAL ANTISYMMETRIC
SHEAR
DEFORMATION
OF RECTANGULAR
PANELS
AND
ANGLEPLY
ON
PLATES,
CYLINDERS
OF
CONSTRUCTION
K. P. SOLDATOS Department of Mathematics, University of loannina, loannina, GR453 32 Greece (Received 9 September 1986, and in revised form 21 February 1987)
This paper is concerned with the influence of thickness shear deformation and rotatory inertia on the free vibrations of antisymmetric angleply laminated circular cylindrical panels. Two kinds of thickness shear deformable shell theories are considered. In the first one, uniformly distributed thickness shear strains through the shell thickness and, therefore, thickness shear correction factors are used. In the second theory a parabolic variation of thickness shear strains and stresses with zero values at the inner and outer shell surfaces is assumed. The analysis is mainly based on Love’s approximations but, for purposes of comparison, Donnell’s shallow shell approximations are also considered. For a simply supported panel, the equations of motion of the aforementioned theories, as well as of the corresponding classical theories, are solved by using Galerkin’s method. For a family of graphiteepoxy angleply laminated plates and circular cylindrical panels, numerical results are obtained, compared and discussed and some interesting conclusions are made regarding the shell theories considered as well as the mathematical method employed. 1. INTRODUCTION
Most of the linear or linearized analyses concerned with antisymmetric angleply laminates [l] are based on classical theories, either for flat plates [2111 or for shells [12, 131 and panels [14, 151 with circular cylindrical configuration. However, as is well known, laminated composite plates and shells exhibit much larger thickness shear effects than do corresponding structures made of a homogeneous isotropic material. Hence, analyses of antisymmetric angleply laminates which take into account thickness shear deformation are already available in the literature but they are concerned with the flat plate configuration only [ 16191. This paper is concerned with the influence of thickness shear deformation and rotatory inertia on the free vibrations of antisymmetric angleply laminated circular cylindrical panels. The choice of the cylindrical panel geometry as a basic configuration has been made because, dependent on the value of a shallowness angle parameter [14], both the flat plate and the circular cylindrical shell configurations can be considered as special cases. For the purpose of the present paper, the equations of motion of two firstapproximation theories are mainly used; both theories can be considered as thickness shear deformable analogs of a classical Lovetype theory. The first theory is motivated by Mindlin’s plate theory [20] and its extensions to the circular cylindrical shell configuration [21281. Since it is based on the assumption of a uniform thickness shear deformation, it is not possible to satisfy the boundary conditions of zero thickness shear stresses at the inner and outer shell surfaces and, therefore, use has to be made of thickness shear correction factors. The second theory, which is motivated by an idea already employed in plates [18,29311 111
0022460X/87/2201 11+ 28 SO3.00/0
@ 1987 Academic Press L.imited
112
K. P. SOLDATOS
and shells [32361, exhibits an advantage in the mathematical description of the physical problem considered; a realistic parabolic variation of thickness shear strains is assumed, which allows zero thickness shear stresses at the extreme fibers of the panel. Its equations of motion include, entirely, the equations of the aforementioned classical Lovetype theory which, therefore, is directly obtained as a special case. It must be noted here that, so far, the problems of buckling and/or vibration of antisymmetric angleply laminated circular cylindrical shells and panels have been studied only on the basis of the classical Donnelltype theory [ 12 151. However, as is well known from corresponding analyses concerned with homogeneous isotropic or crossply laminated thinwalled structures, the accuracy of Donnell approximations is restricted to either short cylindrical shells or shallow panels. Hence, a question regarding the accuracy of the Donnell approximations enters into the problem considered. Therefore, for each one of the three Lovetype theories considered (the two thickness shear deformable theories and the classical one), the corresponding Donnelltype equations are obtained as special cases. For the solution of the free vibration problem, it is assumed that the cylindrical panel considered is subjected to S3 simply supported edge boundary conditions (according to Almroth’s classification [37]). For the classical Donnelltype theory employed in reference [14], a displacement model satisfying exactly this type of boundary conditions was proposed, in a Fourier series form; the solution obtained for the corresponding equations of motion was then based on the application of Galerkin’s method. In the current paper, the same solution procedure is followed for the differential equations of all shell theories employed. In particular, the choice of a Fourier series displacement model suitable for the solution of the equations of any one of the thickness shear deformable theories is based on the already known exact solution obtained for the corresponding flat plate problem [16181. Due to the infinite complexity of the class of antisymmetric angleply laminates, numerical examples concerning a special arrangement of graphiteepoxy laminae only are considered. For the particular case of a flat plate, the results obtained are in fairly good agreement with corresponding results already available in the literature. For open cylindrical panels, further numerical results based on all thickness shear deformable theories considered as well as corresponding classical shell theories are obtained, compared and discussed. For closed cylindrical shells, it appears that the mathematical method employed fails, in the sense that it cannot take into account the inplane and outofplane coupling due to the nonzero shell curvature. 2. FORMULATION OF GOVERNING EQUATIONS Figure 1 shows the nomenclature of the middle surface of a circular cylindrical panel with radius R, axial length L, and circumferential length L,; the constant thickness of
Figure
1. Nomenclature
of a circular
cylindrical
panel.
ANGLEPLY
LAMINATED
CYLINDRICAL.
113
PANELS
the panel is denoted by h. The parameters x, s and z denote length in the axial, circumferential and normal to the middle surface direction, respectively. The panel is an antisymmetric angleply one [l]. As such, it is composed of an even number of layers perfectly bonded together. Each individual layer (say the kth one) is considered to behave macroscopically as a homogeneous, generally orthotropic (monoclinic) linearly elastic material whose state of stress is governed by the Hooke’s law
where the Q’s denote reduced stiffnesses [I, 161. However, due to the properties of an antisymmetric angleply laminate, the arrangement of these layers through the panel thickness is such that symmetry in thicknesses but antisymmetry in material properties exist, with respect to the panel middle surface. In more detail, every two layers that are symmetrically arranged with respect to the middle surface have the same thicknesses and are constructed of the same orthotropic material, but their principal material directions form opposite angles to the panel axis (see for instance reference [l], Tables 4 and 5). 2.1.
A THEORY
EMPLOYING
The following
UNIFORM
displacement
THICKNESS
model
SHEAR
DEFORMATION
is considered:
U(x, S, z; t) = u(x, s; t)+zl//\(X, s; t), V(x, s, z; t) = V(X, s; t) + z&(x, s; t), where t denotes time. According to surface displacement components, the s and x directions, respectively, deformation. By using the displacement field employed here [38,39], the strains extensional and flexural components e, = e, + zk,,
E, = e., +
W(x, S, z; 1) = w(x, s; r),
(2)
Mindlin’s plate theory [20], u, u and w are the middle while r,& and $.Yrepresent the angular rotation, about of straight lines normal to the middle surface before (2), and for the Love’s first approximation theory appearing in equations (1) can be decomposed into according to
zk,,
&XS e,, + zk,, ,
e.Xz= eXZ,
e,z = e PZ,
(3)
where
e, = u,,, e yz
=
e,=v,.,+wlR,
IL,+ w., h,vIR,
ex,=v,,+u,,,
k = kx,
ks= A,.,,
exz=$x+w,x,
km= A., + h,s.
(4)
here, works as a tracer in a manner similar to The Kronecker’s symbol, aLT, introduced that employed in references [28,40]; the subindex T takes the value L for a Lovetype theory, while it takes the value D for a shear deformable analog of the Donnelltype theory used in reference [14]. The force and moment resultants are, respectively, defined as follows: h/2 (N,
NT, N.,,
Qx, 0.7) =
I
(rx, UT, 7x,, 7x,, 7,z) dz,
cm, MT, Mm)
h/7
h/2 =
h/2
(ux,,~7, ~x,)zdz.
(5)
K. P. SOLDATOS
114
By introducing equations (1) into equations (5) and taking into account the geometric and material symmetries of an antisymmetric angleply laminate, the following constitutive equations are obtained:
N, NS NXS = MX MT MX,
A,, Au Al2 h2 0 0 ___
0
I 0
0
0
/
0
0
4, j
0
0
B,,
1 D,,
D,2
0
0
B,,
) D,z
022
46
Bx
0
; 0
0
[ Ox 0, I[ = kA4 0
(6)
(7)
ks& 0 I[ exs e,,1
Here
(A,, Bi,, Di,) =
Joh,_1 Q1:‘U, z, z’) dz,
are, respectively, the wellknown extensional, in classical plate [l] and shell [40] theories,
coupling while
(8)
i,j=1,2,6, and bending
stiffnesses
appearing
h/l
(Am, Ass)= __h,2(Q%‘, Q::‘) dz.
(9)
The constants k, and k, appearing in equations (7) are the analogs of the shear correction factors introduced by Mindlin [20] for the dynamic analysis of homogeneous isotropic plates. Their Mindlin’s value ( 7r2/ 12) have been also found suitable for corresponding analyses of homogeneous circular cylindrical shells [22,23,41,42]. In the case of a laminated composite structure these factors account not only for the variations of shear angles and complex stress state but also for the types of materials, the manner in which they are assembled as well as the geometric characteristics of the particular laminated structural element. Hence, for their calculation, several static and dynamic methods have been developed [25,43451. On the other hand, since there is no “exact” means of determining k, and kg, their Mindlin’s value has been used even in higher order shell theories [46,47]. Hence, except for cases in which something different is noted, Mindlin’s value (k, = k5 = T’/ 12) is adopted here, too. For the thickness shear deformable analog of both the Lovetype and the Donnelltype classical theories, the equations of motion governing the free vibration problem of cylindrical shells can be expressed in the following unified form [28]: N.,,, + N,,., = PoU.rl + P,&,rr,
N,,, + NT.5+
0x.x + Q,,,  N,IR = ~ow,rr,
Here the inertia
terms are defined
&,QJ
R=
~ov,t,
+
p,k,,,
Wx + Mx,,,  Qx = PIu,,, + ~zlClx,rr,
according
to
,I: 2 C’,=
pz’ dz, i
i=o,
1,2;
(11)
hi2
the mass density p may differ, in general, from layer to layer according, study, to the material symmetries of an antisymmetric angleply laminate.
in the present
ANGLEPLY
LAMINATED
CYLINDRICAL
Introduction of expressions (4), (6) and (7) into equations eigenvalue problem of the standard form
[=u~~~= 101,
5=s/Lv
(10) leads to a differential
(121
iv= (4 0,w,wx, L&I,
where [,ie] is a 5 x 5 symmetric matrix of partial version of its components is given in Appendix 77= xl.&,
115
PANELS
differential operators; a nondimensional A, in terms of the following parameters: A = L,/ L.,,
(O~%~~l),
(At,, Bii, Di,) = (A,, B,/‘Lx, D,/‘LZ)IA,,,
(Po F, 6) = (AL:,
4 = Lsl R, P,Lz, d/A,,., (13)
The variable C#Jdefined here and appearing in expressions (Al) is the panel shallowness angle indicated in Figure 1. For 4 = 0 (or equivalently R = a) equations (13) and (Al 1 reduce to the corresponding equations of antisymmetric angleply laminated plates which account for uniform thickness shear deformation [ 16,171. For 4 = 27r and T = L (or Ll), they reduce to the corresponding Lovetype (or Donnelltype) equations of thickness shear deformable antisymmetric angleply laminated shells. 2.2.
A THEORY
EMPLOYING
The following
PARABOLIC
displacement U(x,s,z;
V(x,s,z;
VARIATION
OF THICKNESS
SHEAR
field is considered:
t)=u(x,
s; t)z[w,,(l4z’/3h2)u,(x,s;
t)=(l+&,z/R)u(x,s;
t)],
t)z[w,,(I4z’/3h2)vl(x,s;
t)],
W(x, s, z; t) = w(x, s;t);
(14)
this applies for either a Lovetype (T = L) or a Donnelltype (T = D) shear deformable theory. As in the displacement model (2), u, u and w again represent the shell middle surface displacement components. The terms zw,, and z( w,,  SLTu/ R) are the standard terms which guarantee the validity of the KirchhoffLove assumptions in the classical Lovetype (T = L) or Donnelltype (T = D) theory; evidently, upon neglecting the remainfields (14) reduce to the corresponding fields ing terms (u, = u, = 0), the displacement used for both classical theories. These terms, involving u, and u,, have been employed to disturb the assumption that normals to the undeformed middle surface still remain normal to it after deformation; they also remove the assumption, employed in the displacement model (2), that these normals remain straight after deformation. As can be seen from expressions (15) below, the unknown functions u, and I.+ represent the action of the thickness shear strains on the shell middle surface. Upon introducing expressions (14) into the linear version of the straindisplacement relations of the threedimensional elasticity [39] and restricting approximations to the limits of a first order shell theory (h/R cc l), the strain components appearing in Hooke’s law (1) are obtained in a power series form of the transverse coordinate z as follows: ~,=e,+zk,+(4z~/3h~)f,,
F, = e, + zk, + (4z3/3h2)l,,, E,, = (1  4z2/ h2)e,,,
F,, = ers +zk,, + (4z3/3h2)1,,,
F,; = (1 4zZ/h2)eszr
(15)
where eX = u.,, k.r = ul,,  w,,,,
e, = u,, •t wIR,
e,, = u., + u,,,
k., = VI,,  w,,, + &,u,,lR, !, = ui,r,
1%= %,r,
erz = ul,
erz = uI,
k,, = VI,, + ul,.y  2w,,, + &,v,.JR, 1.X,= (t&X + U1.S).
(16)
116
K. P. SOLDATOS
through the shell Evidently, both thickness shear strains, eXZand E,;, vary parabolically thickness and take on a zero value at the inner and outer shell surfaces. Accordingly, Hooke’s law (1) suggests that although thickness shear stresses may present discontinuities at the interfaces between successive layers, they also have zero values at the extreme fibers of the shell. Hence, the boundary conditions of zero thickness shear stresses at the inner and outer shell surfaces are satisfied. The following force and moment resultants are defined: h/2
h/2 (Ns,
Ns,
Nx,)
(a;,
=
a\,
7x7)
dz,
(4,
Qs)
(7x2,
=
I
r,z)(l
4z2/h2)
dz,
I
(Sx,
ss,
=
Sx.5)
j$
(a;,
y2 i
a,,
7,s)~~
dz.
h/2
(17) Introducing equations (1) into equations material symmetries of an antisymmetric tive equations:
NX’ NT NX, _ MX MS = MX, SX SS _S xs
(17) and taking into account the geometric and angleply laminate, gives the following constitu
0 Lq6 0 i 0 0 B2b 0 j 0 B 0 0 0 AM ’ 4, ___________,______2~____L~2~__
A,, Au
A12 A22
0
0
BK, 1 D,,
0
0
B2e
Bib
BZh
0
0
0
Bib
) D;,
0
0
B&
) D’12
B;6
B&
0
0
0
Bib
0
0
Bib
B’
B’
0
!
)
D,,
fk _.
k k ,
D12
0
LX,
0’12
D,,
0
D’,I
042
0 0
0
Oh6
0
0
D&
Di2
0
D;,
DT2
0
1
OS2
0
Dy2
Di2
0
1,
0
D&,
0
0
DL6
I I
ex e.,
0
0
[:I=FAt AIl[:rl.
(18)
k
x9
I
1
x.5
_
(19)
D, stiffnesses (i,j = 1,2,6), also Here the extensional A,, coupling B,, and bending appearing in classical theories, are defined by expressions (8). The additional coupling, Bk, and bending, D> and Dz, stiffnesses occurring in the present theory are defined as follows:
(B:,,D:,,D;)=~~_~~20::I(z’,z4,~z6)dz,
i,j=l,2,6.
(20)
Apparently, the B”s couple extensional effects with additional bending effects (S’s and I’s) caused by the assumption of the parabolic variation of thickness shear strains. Moreover, the D”s relate classical (M’s and k’s) with additional (S’s and I’s) bending effects, while the D”‘s relate additional moment resultants (S’s) with additional changes of curvatures (I’s). Finally, note that the thickness shear stiffnesses A44 and Ass, appearing in expressions (19), differ from those defined by expressions (7); here they are defined according to 1,)’ 2 (A,,,
Ass)
(Q$,
= h/2
Q::‘)(
1
4z2/h’)*
dz.
(21)
ANGLEPLY
LAMINATED
CYLINDRICAL
117
PANELS
For the thickness shear deformable analog of both the Lovetype and the Donnelltype classical theories, the equations of motion, governing the free vibration problem of cylindrical shells, are derived by using Hamilton’s principle; they can be expressed in the following unified form: X,x + Nx.,, =bou+Pt(% M,, + Nx.,. + (&,lR)(M,,
 w,,) (4/3~‘h~J,t,,
+ Mu,,)
=[Pou+Pl(v,w,,+2SL_,vlR) +(1/~)pz(~,w,,+~~~vIR)(4/3~‘)(p~+p~/~)~,l,,,, (l/R)N,+M,.,+M,.,,+2M,,..s
=[~ow+~,(~,.~+~,,)+p,[~,,,+~,,,+~,,(~,,/R)~~,,,~,,,l +(4/3h2Mu,,,
+
~I,F)l,ll,
(M,S,),,+(M,,S,,),,Q, =rP,u+p,(u,w,,)(4/3h2)p,u+(4/3h2)p,(y,2u,) + 06/9~4h~,l,,,,
(M,S,)..,+(M,,S,,),,
Qs
=[~,~+~z(~,~,,+~~~~lR)(4/3~*)p~~+(4/3~*)~4(w,.~2~,~~.~~/R) (22)
+ 06/9~4h~,l.,,. They are associated
with the following
edge boundary
conditions:
(a) at x = 0, L, : u prescribed w prescribed u, prescribed
or N, = 0,
v prescribed
or M,,, + M,,, = 0, or M,  S, = 0,
or NY, + 6,,M,,/ w,, prescribed
u, prescribed
R = 0,
or M, = 0,
or M,,  S,, = 0;
(b) at s = 0, L,: u prescribed tvprescribed u, prescribed
or N,, = 0,
v prescribed
or M,,, + M,,,, = 0, or M,,  S,,, = 0,
or N, + SLTM,y/ R = 0,
w,, prescribed v, prescribed
or M, = 0,
or M,  S, = 0.
(23)
The inertia terms appearing in equations (22) are still given according to the definition (ll), where now il,2 ,..., 6. The introduction of expressions (16), (18) and (19) into equations (22) leads again to the standard differential eigenvalue problem
[ms) = (01,
{F= lu, 4 N’Lu,, , L%I,
where [p] is a 5 x 5 symmetric matrix of partial version of its components is given in Appendix
(24)
differential operators; a nondimensional A, in terms of the parameters defined by
K. P. SOLDATOS
118 expressions
(13) as well as the following
(B&,D~,D~)=(BI,/L,,D:,/L~,
parameters:
DQ/L:)/A,,,
(/%,PzJ = (~3, dLs)L.Jh2A,,, (25)
&=dh4A,,.
It is of interest to point out that upon neglecting the contribution of all terms involving the unknown functions u,, and v,, expressions (15) and (16) reduce to the straindisplacement relations of the classical Donnell or Lovetype theory (for T = D or T = L, respectively). Moreover, equations (18) and (22) reduce to the constitutive equations and equations of motion, respectively, of the corresponding classical theories, provided that p. are retained in equations (22). Hence, the differential only inertia terms containing eigenvalue problem corresponding to both classical theories is obtained in the form
ms”~ = {OI, where [Z] is a 3 x 3 symmetric expressions (A2) only inertia 1,2,3) becomes identical with T = D, each one of $, reduces theory presented in reference
3. SOLUTION
(26)
{6”IT = (4 v, WI,
matrix of partial differential operators. Upon retaining in terms containing PO, each one of the ope_rators _5??,,(i, j = the corresponding one of the operators 9. Moreover, for to the corresponding operator of the classical Donnelltype [ 141.
FOR S3 SIMPLY
SUPPORTED
PANELS
It is now assumed that the cylindrical pane1 considered is subjected to S3 simply supported edge boundary conditions, at its all four edges. That is: (a) for the theories employing uniform thickness shear deformation (section 2.1): at n=O,l:
u=w=+~=O,
(274
at .$ = 0,l:
v = w = 4X = 0,
(27b)
(b) for the theories at n=O,l:
employing
parabolic
thickness
shear deformation
(section
u=w=v,=O,
N,,+(S,,/R)Mx,
=A,,(Av,,+u.,)+AL.~(B,,B;,)u,,, &wm,
+ LA& %jh,t A‘&dw,gt ++#@,6~,rl
+A ‘
[email protected])
&+,E) X’&W,~(
+Lx(D66Dk6)(v,,rl+A‘~,,~)+A~‘~D66~,~]=0,
2.2):
ANGLEPLY
at [=O,l:
LAMINATED
CYLINDRICAL
PANELS
Nxs & I[
v=w=u,=O, MS
=
[ s.5
&,
G6
For the free vibration
problem,
w=cos(wt)
the following
displacement
model
is considered:
F f c,,sin(m7rn)sin(nrr[). m=, nz,
(29)
Here w represents a certain unknown natural frequency, m and n are the axial and circumferential halfwave numbers, respectively, and a,,,,,, b,,, c,,, d,,, D,,, emn and E,, are unknown constant coefficients. The displacement model (29) satisfies exactly the boundary conditions (27). It further satisfies exactly the boundary conditions (28) but only for the case of the Donnelltype theory (T = II). For T = L (Lovetype theory), it is unable to satisfy the natural boundary conditions but, among the conditions (28), at least satisfies the displacement ones. Accordingly, the Fouriertype expansions (29) form an appropriate displacement model which, through the application of Galerkin’s method [ 14, 15,481, may lead to an approximate solution for both differential eigenvalue problems (12) and (24): that is, for all four shear deformable shell theories presented in section 2.
In more detail, Galerkin’s method leads to the following set of equations:
1 1 [~,,u+~,~v+~,~w+L~~,~((C~~,~,)+L~15(k, v,)lsin (WI) cos(M)dv d5 51(1 0 
’[N,,(v, 0) (l)‘N,,(rl, 1)l sin (bv) dv = 0, I0
’
u,)+L,~~~(~G,
[~2,u+~22v+FLa23W+Lr~2J(~r,
K(‘4 O+;
Mx,(O,O(l)
v,)lcos (jv) sin CM3 dv d5
Nx,(l, 5)+i M.x.,(l,5)
II
sin (ir[)
d.$ = 0,
120
K. P. SOLDATOS

’ [M,(O,
I
0
i
0
sin (MI 4
5)(1YW1,5)l(j~)
‘[M,(s,O)(1)‘M,(t),I)](i~)sin(jn_r))dl)=O.

1[~~1~+~~2~+~~~~+~~~~3(~.~,~I)+~r6P4S(~~,~~)1cos(j~~)sin(i~5)drldS 0 0 ’
II

I
1
JI
‘{M,(O,5)S,(O,5)(1)‘[M,(1,5)S,11,5)1}sin(i~5)d5=0, 0
1
0 0 
[~~,u+~~,u+~~,w+L,~~,(cL,,
I
~~11 sin (jm) ~0s(id d5
ul)+L%dh,
‘{M,(q,O)S,(q,O)(l)‘[M,(?,
~)S,(T, l)l)sin (jv)dv=O.
(30)
0
Here il,2 ,..., N and j=1,2 eigenvalue problem of the form Ti,
TG
T,,
T,4
T,s
TZi
T22
T>3
T>4
TX
T3, T 41
TX
T,,
T,,
TX
7.42
T4,
T44
T45
,...,
M. Equations
Hi, w7
0
(30) lead
Hi,
to a general
H,4
0
algebraic
A
0
H?>
HX
0
HX
B
Hi,
HX
H33
H,,
HXS
C ={O],
H43
H44
H41
0
0
Hz2
i.’
+,W’I~[
0
0
D
Tsz T,, T54 Ts I Ts, _I:: E where 0 represents a proper zero matrix. The components of the MN x MN square matrices T, and H, as well as the column matrices A, B, C, D and E are given in Appendix B. It must be noted that, since expressions (29) satisfy exactly the boundary conditions (27), the line integrals appearing in eqatuions (30) do not contribute to the final solution (31), for either one of the theories employing a uniform thickness shear deformation (see expressions (Bl)). Moreover, they do not contribute even in the case of the Donnelltype theory employing a parabolically distributed thickness shear (see expressions (B5), for model (29) satisfies exactly the whole T = D); this is because, for T = D, the displacement set of the boundary conditions (28). For both Donnelltype and Lovetype classical shell theories, the solution of equations (26) can similarly be obtained in the form
[[ ;;I
H,,
(31)
i]=iol,
Hz
(32)
as a particular case of the solution (31) obtained for the corresponding shear deformable shell theories in which a parabolic variation of thickness shear is assumed. Hence, in the algebraic eigenvalue problem (32), the components of the MN x MN square matrices T, (i,j = 1,2,3) are given by expressions (B6), I is a proper unit matrix and A, B and C denote the column matrices defined in expressions (B4). For T= D (Donnelltype theory), the solution (32) should give identical results with the corresponding solution obtained in reference [ 141. However, a comparison of relations (B2) with the corresponding relations (16) in reference [ 141 shows that the submatrices corresponding to T, in reference [ 141 have erroneously been arranged in their transposed form. Therefore, some of the numerical results concerned with cylindrical panels in
ANGLEPLY
LAMINATED
CYLINDRICAL
PANELS
121
reference [ 141 are considerably inaccurate, especially for large values of the shallowness angle 4. However, it must be noted that all numerical results concerned with flat plates ( $J = 0) [ 141 or circular cylindrical shells (4 = 27~) [ 131 are correct since, in both cases, each one of the submatrices T, results in a diagonal form: that is, a symmetric form. In what follows an attempt is made to present the correct results for all 4 # 0 cases treated in reference [14]. Moreover, further results based on both classical as well as all four shear deformable theories are presented and compared, and the influence of thickness shear deformation on vibration of antisymmetric angleply laminates is discussed.
4. NUMERICAL
EXAMPLES
Since there is an infinite complexity of the class of antisymmetric angleply laminates, numerical examples concerning “regular” antisymmetric angleply laminates only are considered: that is, laminates composed of an even number of equalinthickness layers constructed of the same orthotropic material and with principle material directions alternately oriented at angles +B and 0 to the panel axis. For this kind of laminated composites, it can be shown that the only nonvanishing coupling stiffnesses, B,,, Bzh, to the shell’s number of layers. Thus, the coupling B& and B&, are inversely proportional between bending and extension due to the panel lamination dies out rapidly as the number of layers increases and, therefore, a panel composed of an infinite number of layers behaves, macroscopically, like one constructed from a homogeneous orthotropic material. 4.1.
FLAT
PLATE
RESULTS
(4
= 0)
All terms involving aL7 in expressions (Al) and (A2) are also multiplied by 4 and. therefore, for 4 = 0, they disappear. Accordingly, in the particular case of a flat plate, there is no need to distinguish between a Donnelltype and a corresponding Lovetype theory. Similarly, all terms involving 4 disappear in expressions (Bl) and (B5) and among them all terms containing 0 factors. Hence, all submatrices T, and H, (i,j = 1,2, . ,5) appearing in equation (31) are obtained in diagonal forms. As a result, both main matrices constituting the eigenvalue problem (31) can be written in block diagonal forms with, a total of MN, 5 x 5 square submatrices in their main diagonal. Under this special arrangement, the original eigenvalue problem (31) is reduced to the solution of MN eigenvalue problems of the form (31), where T, and H,, represent the individual elements of each one of the 5 x 5 submatrices. These elements are still given by expressions (Bl) or (B5) for the shear deformable theory with either a uniform or a parabolic thickness shear deformation, respectively (in both cases, the indices r and s have no meaning). Similarly, for 4 = 0, the original eigenvalue problem (32) is reduced to the solution of MN eigenvalue problems of the form (32), where T,, (i, j = 1,2,3) represent individual elements of 3 x3 matrices. Again there is no need to distinguish between a classical Donnelltype or Lovetype theory. In both cases the solution (32) coincides with the corresponding classical plate solution obtained in reference [ 141 and characterized as an exact one. All flat plate results presented in reference [14] have shown a fairly good agreement with corresponding results based on the classical plate solution previously obtained, in references [5,8]. For 4 = 0, both shear deformation solutions (31) may be characterized as exact ones, in the sense described in reference [14] for the solution of the classical plate theory (CPT). In particular, for 4 = 0, the solution obtained for the theory employing a uniform shear deformation (USDT) coincide with the corresponding solution obtained in reference [17]. On the other hand, the solution obtained for the theory employing a parabolic variation of shear deformation (PSDT) is in fairly good agreement with the corresponding
122
K. P. SOLDATOS
solution obtained in reference [ 181. The numerical results presented in Tables l3 confirm these observations. Table 1 shows the value of the lowest nondimensional frequency parameter w* = wLf,(pJ E2h3)1’2,
(33)
obtained for several values of the wave numbers m and n of a square plate. The plate, having a thickness to length ratio h/L, = 0.1, consists of four graphiteepoxy layers having the following material properties: E,/E2=40r
G,,/ Ez = G,,/ E, = 0.6, TABLE
b’= 0.25;
GA E2 = 0.5,
(34)
1
Lowest frequency parameters o* of a fourlayered graphiteepoxy (4=0’, h/L,=O.l, 6=45”)
square plate
m
Plate theory
n 1
_
Reference [ 171
1
2
3
4
5
18.46 18.42 18.32
34.87 34.75 34.56
54.27 54.04
97.56 97.05 97.75
53.87
75.58 75.21 75.30
2
Reference USDT PSDT
[ 171
34.87 34.75 34.56
50.52 50.29 50.13
67.17 66.83 66.97
85.27 84.81 85.57
104.95 104.37 106.09
3
Reference USDT PSDT
[ 171
54.27 54.04 53.87
67.17 66.83 66.97
82.84 82.38 83.15
99.02 98.46 100.27
116.28 115.60 118.89
4
Reference USDT PSDT
[ 171
75.58 75.21 75.30
85.27 84.81 85.57
99.02 98.46 100.27
114.45 113.77 117.10
130.31 129.53 134.89
5
Reference USDT PSDT
[ 171
97.56 97.05 97.75
104.95 104.37 106.09
116.28 115.60 118.89
130.31 129.53 134.89
145.57 144.68 152.67
USDT PSDT
TABLE
2
Fundamental frequency parameters wz,,, of twolayered graphiteepoxy (f#J= O”, I9 = 45”) USDT
CPT Reference
h1L.x 0.5 0.25 0.2 0.1 0.05 0.04 0.02 0.01
[I81 6.283 12.566 13.885 14.439 14,587 14.605 14.630 14.636
square plates
PSDT
Reference Present 6.883 13.766 14.315 14.556 14.617 14.625 14.634 14.637
Reference
[I81 k,=k,=n=/12 5.520 9.168 10.335 13.044 14.179 14.338 14.561 14.618
5.491 9.127 10.304 13.028 14.173 14.335 14.560 14.618
k,=kg=s 5.521 9.161 10.335 13.044 14.179 14.338 14.561 14.618
1181 6.283 9.759 10.840 13.263 14.246 14.383 14.572 14.621
Present 6.337 9.759 10.840 13.263 14.246 14.383 14.572 14.621
ANGLEPLY LAMINATED CYLINDRICAL PANELS
123
TABLE 3 Fundamental
frequency
parameters
W*,~,, of eightlayered graphiteepoxy (4 = O”, f3= 45”)
CPT
USDT
Reference
[181
Present
0.5
6283 12.566 15.708 25.052 25.212 25.232 25.258 25.264
6.883 13.766 17.207 25.256 25.264 25.265 25.266 25.266
0.25 0.2 0.1 0.05 0.04 0.02 0.01
PSDT .F
Reference
h/L
[I81
5.848 10.842 12.892 19.289 23.259 23.924 24.909 25.176
k, = k5 = rr’f 12 k, = k, = 2
5.813 10.785 12.831 19.238 23.236 23.908 24.905 25.175
square plates
5,848 IO.842 12.892 19.289 23.259 23.924 24,909 25.176
Reference
[I81 6.283 10.991 12.972 19.266 23.239 23.909 24.905 25.174
Present 6.314 10.991 12.972 19.266 23.239 23.909 24.905 25.174
the lamination angle is 0 = 45”. Table 1 demonstrates the good agreement of numerical results based on USDT and PSDT with corresponding results cited in Table 2 of reference [17]. The small discrepancies observed between corresponding results based on USDT and the uniform shear theory used in reference [17] are due to the value used here for the shear correction factors (k, = k5 = rr’/12). In fact, by using the value employed in reference [17] (k4= k5 = 5/6), all results obtained by using USDT were identical with the corresponding results presented in reference [ 171. Tables 2 and 3 show the fundamental frequency parameter W~in of a twolayered and an eightlayered square plate, respectively, obtained for several values of the thickness to length ratio. Graphiteepoxy layers having the material properties (34) are considered; the lamination angle is 0 = 45”. Numerical results based on both shear deformable theories as well as the classical plate theory are presented and compared with corresponding results cited in Table 4 of reference [18]. From the results presented in both Tables 2 and 3 the following observations can be made. (1) Except for thick plate cases (h/L, 2 O2), all results based on CPT are in very good agreement with corresponding results based on the classical theory used in reference [ 181. The observation that corresponding results do not coincide, as well as that for h/L,. 2 0.2 considerable discrepancies occur, cannot be investigated since, in reference [lg], no information is given about the classical plate theory employed. (2) For k4 = k, = 5/6, all results based on USDT are identical with the corresponding results based on the uniform shear theory used in reference [18]; for k, = k, = &/12, there is a very good agreement. (3) Except for the case that h/L, = 0.5, all results based on PSDT are identical with the corresponding results based on the parabolic shear theory presented in reference [ 181. 4.2. CIRCULAR CYLINDRICAL SHELL RESULTS (4 = 27r) A closed circular cylindrical shell (4 = 2~) vibrates in circumferential fullwaves and, therefore, n and i must be even integers. Since in this case O(n, i) = O(i, n) = 0, all terms involving 0 factors disappear. Hence, as in the flat plate case, the original eigenvalue problem (31) is reduced to the solution of MN eigenvalue problems, where T,, and H,, 5) represent individual elements of 5 x 5 matrices. Similarly, the eigenvalue (i,j=l,2 ,..., problem (32) is reduced to the solution of MN eigenvalue problems, where T,, (i, j = 1,2$3) represent individual elements of 3 x 3 matrices. However, as has already been mentioned for the classical Donnelltype theory [ 13, 141, the solutions obtained are not the exact
K. P.
124
SOLDATOS
ones. As is explained next, the disappearing terms do not contribute to the final solution, not because of the particular shell configuration, as happens in the case of the flat plate, but due to the failure of the mathematical method employed. In fact, in the particular case of a shell composed of an infinite number of layers (B,, = B,, = Bi6 = B&, = O), both eigenvalue problems (31) and (32) are considerably simplified. In more detail, they are obtained in the form 2 T,,
[email protected]
HI,
7,2
;
0
T22co2H22; 0 TX __________,_____2_____0 0 ITCW Hu 0 0 ; T4, 0
0
;
T44

0
0
A
0
0
B __
T 34
TS5
c
T4.5
D
~‘H44
T 54
Tsx
and T,, P&J2 T2, 0 c
T12
;
0
1 TrPoo’
IC I
(35)
E
A B
T22Pow’;___O____
0
Tssw2HSs
={O}
={O>,
(36)
C
respectively. Apparently, the approximate solution (36) of the classical shell theories states erroneously that, as in the classical plate theory, there is no coupling between inplane and flexural modes of vibration. Moreover, since the value of T,, is independent of whether the Donnelltype or the Lovetype theory is used, the approximate solution (36) gives erroneously identical flexural frequencies of vibration for both classical theories. In a similar manner, the approximate solution (35) of all thickness shear deformable theories states erroneously that, as in corresponding plate theories, there is no coupling between inplane and outofplane (flexural and thickness shear) modes of vibration. Moreover, since the expressions for Tj and H,, (i, j = 3,4,5) are independent of ~?~r, the approximate solution (35) yields erroneously that (1) identical outofplane frequencies are obtained by using either one of the two shear deformable theories and assuming a uniform thickness shear deformation (section 2.1), and (2) identical outofplane frequencies are obtained by using either one of the two shear deformable theories and assuming a parabolic variation of thickness shear deformation (section (2.2). Hence, it appears erroneously that for a circular cylindrical shell composed of an infinite number of layers, one has to distinguish between Donnelltype and Lovetype approximations, only if one is interested in inplane modes of vibration. In the case of a shell composed of a finite number of layers, the coupling stiffnesses B,6, B26r Bib and B& do not vanish. Therefore, further simplifications of the eigenvalue problems (31) and (32), analogous to those made for the infinitelayered shell, are not possible. Accordingly, although one has always to distinguish between Donnelltype and Lovetype approximations, the mathematical method employed still fails, in the sense that it still “ignores” the inplane and outofplane coupling due to the nonzero shell curvature; it takes into account only the corresponding coupling due to the shell lamination. These observations suggest that one should postpone discussion of numerical results concerned with circular cylindrical shells (4 = 2~7). Such a discussion might be found of interest in connection with corresponding solutions obtained on the basis of some other mathematical methods (see, for instance, references [49, SO]). 4.3.
CIRCULAR
CYLINDRICAL
PANEL
RESULTS
(c$ # 0,2?r)
As has already been mentioned, some of the numerical results concerned with cylindrical panels in reference [14] are considerably inaccurate. In this section the correct results
ANGLEPLY
LAMINATED
CYLINDRICAL
125
PANELS
are presented for the classical Donnelltype theory used in reference [14] while, at the same time, some information is given about the influence of thickness shear deformation on the free vibration of regular antisymmetric angleply laminated circular cylindrical panels. In view of this consideration, and due to the large value of parameters entered in the problem, attention will be restricted to the family of graphiteepoxy panels used in reference [14]. These panels have the same axial length (L, = b), thickness (h/b ==0.05) and plan form (b x b), so that upon changing the value of the shallowness angle 4, the panel circumferential length changes according to L, = bc$/2 sin (4/2).
The normalized fundamental
(37)
frequency parameter W= w,i,b2(po/ E2h3)“‘,
(38)
as a function of the lamination angle 0, is shown in Figures 24 for several values of the shallowness angle $J and for various numbers of layers. All results drawn with a dashed line are based on the Lovetype CST theory while solid lines represent results based on the Lovetype PSDT theory. The numerical values of some of the frequencies drawn in Figures 24 are tabulated in Table 4, together with corresponding results based on the Lovetype USDT. Moreover, corresponding results based on the Donnelltype theories are tabulated in Table 5. The results cited in Tables 4 and 5 lead to the following 30
1
10 
2 layers
<'
<’
30 
2 layers ‘3
lo
1 2 ibyers
6 (degrees)
Figure 2. Fundamental frequency parameter G as a function of the lamination angle 0 (4 = O”, IO”, 20”). Graphiteepoxy: E, = 40 E,; G,, = G,, = G,, = 0.5 E,; viz = 0.25.   , Classical theory; , thickness shear theory.
126
K. P. SOLDATOS
(m,n)=(2,11
60
50
13 40
30
20 2 layers 10
I
15
1
I
30
45
I
60
I
75
90
8 (degrees) Figure 3. Fundamental frequency parameters and key as Figure 2.
parameter
W as a function
of the lamination
angle
0 (C#J= 30”). Material
r I
/ 70
60
‘3 50
40
30
2c I
0
15
I
I
30
8 Figure 4. Fundamental frequency parameters and key as Figure 2.
parameter
45 (degrees)
W as a function
I
60
I
75
of the lamination
90 angle
19 (6 =45”).
Material
ANGLEPLY
LAMINATED
CYLINDRICAL
TABLE
4
Fundamental frequency parameters (I, of graphiteepoxy Lovetype theories Two layers 4 (degrees)
127
PANELS
circular cylindrical panels based on
colayers
Four layers ,
0
P.
(degrees)
CST
USDT
PSDT
CST
USDT
PSDT
CST
USDT
0
0 30 45 60 90
18.81 14.18 14.62 14.18 18.81
17.17 13.71 14.13 13.71 17.17
17.19 13.78 14.21 13.78 17.19
18.81 22.17 23.52 22.17 18.81
17.17 22.17 21.71 22.17 17.17
17.19 20.49 21.66 20.49 17.19
18.81 24.26 25.82 24.26 18.81
17.17 22.16 23.48 22.16 17.17
PSDT 17.19 22.19 23.52 22.19 17.19
20
0 30 45 60 90
20.01 19.11 26.90 35.59 45.68
18.48 18.79 26.70 35.49 44.42
18.50 18.80 26.65 35.37 4443
20.01 25.59 32.58 39.48 45.68
18.48 24.19 31.31 38.57 44.42
18.50 24.16 31.26 38.53 44.43
20.01 27.41 34.27 40.69 45.68
18.48 25.59 32.56 39.46 44.42
18.50 25.6 1 32.57 39.47 44.43
30
0 30 45 60 90
21.39 23.78 36.69 50.52 62.76
19.96 23.56 36.61 50.27 55.78
19.98 23.52 36.48 50.21 56.34
21.39 29.22 41.02 53.45 62.76
19.96 28.03 40.03 52.74 55.78
19.98 27.99 40.01 52.69 56.34
21.39 30.83 42.37 54.36 62.76
19.96 29.23 40.00 53.39 55.78
19.98 29.26 41.05 53.40 56.34
45
0 30 45 60 90
24.10 31.57 51.79 57.14 70.98
22.83 31.47 52.10 53.62 55.20
22.85 31.36 51.59 53.97 56.08
24.10 35.83 54.97 75.03 70.98
22.83 34.88 54.25 67.73 55.20
22.85 34.84 54.18 67.95 56.08
24.10 37.14 55.98 75.74 70.98
22.83 35.85 54.95 70.51 55.20
22.85 35.86 54.97 71.41 56.08
observations, which explain the reasons why only frequencies based on Lovetype CST fundamental frequencies based on and PSDT appear in Figures 24: (1) corresponding the two CST theories (Donnelltype and Lovetype) are in such close agreement that they cannot be distinguished in drawings; (2) corresponding fundamental frequencies based on the two USDT theories are in such close agreement that they cannot be distinguished in drawings; (3) corresponding fundamental frequencies based on the two PSDT theories are in such close agreement that they cannot be distinguished in drawings; (4) all PSDT results are in good agreement with corresponding USDT results; therefore, only USDT results have been drawn in Figures 24. All fundamental frequency parameters presented in Figure 2 have nominal halfwave numbers (m, n) = (1, 1). For $J = 0 (square plate), the results presented were easily obtained by using the simplifications explained in section 4.1. For 4 # 0, the integers M and N appearing in expressions (29) were gradually increased until a sufficiently good approximation to the actual panel frequencies could be ensured by observation of the convergence of the numerical results obtained. This technique is shown in Table 6 where, for a twolayered panel with 4 = 20” and 0 = 45”, the influence of M and N is indicated on the frequency parameter 6 obtained for all six shell theories. Evidently, all fundamental frequencies predicted on the basis of a thickness shear deformation theory have always been lower than the ones based on a corresponding classical theory. However, the fundamental frequency reduction due to the consideration of thickness shear deformation seems to be dependent on the value of several parameters such as the shallowness angle, 4, the lamination angle, 6, and the number of layers. For
128
K. P. SOLDATOS TABLE 5
Fundamental frequency parameters (3 of graphiteepoxy circular cylindrical panels based on Donnelltype theories Two layers ti (degrees)
Four layers
colayers
6 (degrees)
CST
USDT
PSDT
CST
USDT
PSDT
CST
USDT
PSDT
0
0 30 45 60 90
18.81 14.18 14.62 14.18 18.81
17.17 13.71 14.13 13.71 17.17
17.19 13.78 14.21 13.78 17.19
18.81 22.17 23.52 22.17 18.81
17.17 20.53 21.71 20.53 17.17
17.19 20.49 21.66 20.49 17.19
18.81 24.26 25.82 24.26 18.81
17.17 22.16 23.48 22.16 17.17
17.19 22.19 23.52 22.19 17.19
20
0 30 45 60 90
20.01 19.13 26.93 35.61 45,73
18.48 18.79 26.70 35.49 44.45
18.50 18.82 26.68 35.40 44.47
20.01 25.59 32.59 39.49 45.73
18.48 24.19 31.31 38.57 44.45
18.50 24.16 31.27 38.54 44.47
20.01 27.41 34.28 40.70 45.73
18.48 25.59 32.56 39.46 44.45
18.50 25.61 32.58 39.48 44.47
30
0 30 45 60 90
21.39 23.82 36.74 50.58 62.84
19.96 23.56 36.61 50.38 56.12
19.98 23.56 36.53 50.31 56.37
21.39 29.23 41.04 53.48 62.84
19.96 28.03 40.03 52.75 56.12
19.98 27.99 40.00 52.71 56.37
21.39 30.83 42.38 54.38 62.84
19.96 29.23 41.00 53.40 56.12
19.98 29.25 41.02 53.41 56.37
45
0 30 45 60 90
24.10 31.63 51.92 57.50 71.32
22.84 31.47 52.08 54.05 55.91
22.85 31.42 51.73 54.16 56.14
24.10 35.85 55.01 75.09 71.32
22.84 34.88 54.26 68.36 55.91
22.85 34.85 54.22 68.02 56.14
24.10 37.14 56.00 75.77 71.32
22.84 35.85 54.96 71.18 55.91
22.85 35.86 54.97 71.44 56.14
instance, for small shallowness angles, such as the ones used for the results drawn in Figure 2, the aforementioned reduction is, in general, small and very rarely exceeds the value of the admissible engineering error (5%). On the contrary, upon increasing the value of the shallowness angle (Figures 3 and 4) there are cases in which this reduction is increased considerably. In particular, this hapens for large values of the lamination angle (0 > 70” for 4 = 30”, and 13> 50” for 4 = 45”) where the mode shape changes and the fundamental frequencies occur for combinations of halfwave numbers different from (m, n) = (1, 1) (see Figures 3 and 4). All results drawn in Figures 24 make apparent the wellknown effect of the coupling stiffnesses Br6, BZ6, Bi6 and B&: that is, reduction of the frequency parameters from their orthotropic values (colayered panels). This reduction is more clearly indicated in Figures 5 and 6 where the fundamental frequencies based on PSDT, of two and fourlayered panels, are normalized by the fundamental frequency w0 of the corresponding orthotropic panels ( Br6 = &, = & = Bi6 = 0). The solid lines shown in both figures indicate the flat plate reduction and, as in Figure 2, they are symmetric about 13= 45”. The frequency reductions for twolayered plates with lamination angles 30”, 45” and 75” are about 38, 41 and 26 per cent, respectively, but they are diminished in cases of twolayered panels with shallowness angles 10” and 20”; they become about 35, 31 and 12 per cent and 21, 17 and 6 per cent, respectively. For fourlayered plates the corresponding reductions are about 7, 7.5 and 5 per cent but for fourlayered panels with shallowness angles 10” and 20” are, respectively, about 6.5, 6 and 2.5 per cent and 6, 3.5 and 1 per cent. ’ The effect of the coupling stiffnesses B,6r Bz6, Bi6 and B& seems to be somewhat complicated for panels with 4 = 30” and 45” (Figure 6), especially for large values of the
ANGLEPLY
LAMINATED
CYLINDRICAL
TABLE
129
PANELS
6
of M and Non thefundamentalfrequencyparameterw of a graphiteepox) circular cylindrical panel ( C#J = 20”, 8 = 45”, two layers)
Influence
N
M
I
2
3
4
5 _____
CST Donnelltype
1 2 3 4 5
27.120 27.120 27.120 27.120 27.120
27.120 26.954 26.952 26.944 26.944
27.120 26.953 26.951 26.942 26.942
27.120 26.941 26.939 26.930 26.930
27.120 26.941 26.939 26.930 26.930
CST Lovetype
1 2 3 4 5
27.094 27.094 27.094 27.094 27.094
27.094 26.924 26.923 26.914 26.914
27.094 26.922 26.920 26.912 26.912
27.094 26.910 26.908 26.900 26.899
27.094 26.910 26.908 26.899 26.899
1 2 3 4 5
26.835 26,835 26.835 26.835 26.835
26.835 26.713 26.712 26.710 26.710
26.835 26.714 26.712 26.710 26.709
26.835 26.703 26.701 26.699 26.699
26.835 26.703 26.701 26.699 26.698
1 2 3 4 s
26.835 26.835 26.835 26.835 26.835
26.835 26.713 26.712 26.709 26.709
26.835 26.714 26.711 26.709 26.709
26.835 26.702 26.701 26.698 26.698
26.835 26.702 26.700 26.698 26.698
1 2 3 4 5
26.876 26.876 26.876 26.876 26.876
26.876 26.709 26.707 26.698 26.698
26.876 26.707 26.704 26.695 26.695
26.876 26.695 26.693 26.684 26.683
26.867 26.695 26.692 26.683 26,683
1
26.850 26.850 26.850 26.850 26.850
26.850 26.680 26.678 26.670 26.670
26.850 26.678 26.675 26.667 26.667
26.850 26.666 26.664 26.655 26.654
26,850 26.666 26,663 26,655 26.654
Theory
USDT Donnelltype
USDT Lovetype
PSDT Donnelltype
PSDT Lovetype
lamination mode
2 3 4 5
angle
the reduction becomes respectively, increasing
where
the
fundamental
((m, n) # (1,l)).
shapes
of the fundamental
about
19 and
frequency
11 per cent,
for
&J= 45”. On
d the
aforementioned
frequencies
For instance,
the
cent for panels with 4 = 30” and be made for fourlayered panels.
are
for twolayered continuously
respectively,
associated panels decreases
with with
with increasing
for $J = 30” and about
contrary,
for
twolayered
reduction
increases;
45”, respectively.
panels
it becomes
Corresponding
complicated
8 = 30” and
45” 4; it
12 and 6 per cent, with
about
0 = 75”, upon 13 and
observations
20 per can also
5. CONCLUSIONS The free vibration problem of antisymmetric angleply laminated circular cylindrical shells and panels has been considered and studied, mainly on the basis of Love’s approximations. In detail, the equations of motion of two firstapproximation thickness
130
K. P. SOLDATOS
I
30
60
45
/
75
C
8 (degrees) Figure 5. Fundamental (#)=O”; ___( $=lO”;..,
frequency &:20
reduction
as a function
of the lamination
angle
0 (4 = O”, lo”, 20”). .
3” 0 6  of layers ‘; o4 1 t+<&++ ‘,‘+l 0
15
30
I
1
45
60
I
75
8 (degrees) Figure 6. Fundamental $f)= 0”; _ _ _, 4 = 30”; 
frequency 
reduction
, 4 = 45”.
as a function
of the lamination
angle
0 ( C$= O”, 30”, 45”). ,
shear deformable shell theories have been derived and used for the problem considered. The first theory is an extension of Mindlin’s plate theory; a uniform thickness shear deformation is assumed and, therefore, it involves thickness shear correction factors. In the second theory the use of any shear correction factors is avoided by assuming a parabolic variation of thickness shear deformation; upon neglecting the effects of thickness shear deformation, it leads, directly, to the equations of motion of a classical Lovetype theory. As a special case of each of the aforementioned Lovetype theories, the equations of motion of a corresponding Donnelltype shallow shell theory are obtained for comparison purposes. For a simply supported panel, the differential equations for all six shell theories as mentioned have been solved by using Galerkin’s method. Depending on the value of a shallowness angle parameter, the obtained solution produces, as special cases, the already known exact solution for the corresponding flat plate problem as well as an approximate solution for the corresponding problem of closed circular cylindrical shells. From the
ANGLEPLY
LAMINATED
CYLINDRICAL
PANELS
131
numerical results presented for graphiteepoxy regural antisymmetric angleply laminates, the following principal conclusions can be noted. (1) Flat plate results are, in general, in an excellent agreement with corresponding results already available in the literature. (2) The effect of thickness shear is to decrease the value of natural frequencies of vibration predicted on the basis of classical plate or shell theories. (3) In the particular case of a closed circular cylindrical shell, the mathematical method employed leads to an approximate solution in which inplane and outofplane coupling due to the nonzero shell curvature is ignored. As a result, corresponding Donnelltype and Lovetype theories produce numerical results which are in a fairly good agreement or even identical (for colayered shells). This unexpected observation, especially for long shells, suggests that some other mathematical method might be found preferable for the vibration analysis of closed cylindrical shells. (4) In the case of an open cylindrical panel the mathematical method employed leads to a solution which takes into consideration the inplane and outofplane coupling due to both the nonzero shell curvature and shell lamination. Nevertheless, corresponding Donnelltype and Lovetype theories produce fundamental frequencies which are still in good agreement but, in this case, this must be attributed to the shallowness of the panels considered for numerical applications. (5) In the case of an open cylindrical panel, the reduction of the frequencies of vibration due to the consideration of tb.ickness shear is dependent on several parameters, mainly the shallowness angle, the lamination angle and the number of layers.
REFERENCES 1. R. M. JONES 1975 Mechanics ofComposite Materials. New York: McGrawHill. and Y. STAVSKY 1961 American Society of Mechanical Engineers Journal of Applied Mechanics 28, 402408. Bending and stretching of certain types of heterogeneous
2. E. REISSNER
aeolotropic elastic plates. 3. J. M. WHITNEY 1969 Journal qf Composite Materials 3, 2028. Bendingextensional 4. 5.
6. 7. 8.
9. 10. 11.
12. 13. 14.
coupling in laminated plates under transverse loading. bending of J. M. WHITNEY 1969 Journal of Composite Materials 3, 715719. Cylindrical unsymmetrically laminated plates. J. M. WHITNEY and A. W. LEISSA 1969 American Society of Mechanical Engineers Journal c?f’ Applied Mechanics 36, 261266. Analysis of heterogeneous anisotropic plates. J. M. WHITNEY and A. W. LEISSA 1970 American Institute of Aeronautics and Astronautics Journal 8, 2833. Analysis of a simply supported laminated anisogropic rectangular plate. J. M. WHITNEY 1970 Journal of Composite Materials 4, 192203. The effect of boundary conditions on the response of laminated composites. R. M. JONES, H. S. MORGAN and J. M. WHITNEY 1973 American Society of MechanicaL Engineers Journal of Applied Mechanics 40, 1143l 144. Buckling and vibration of antisymmetritally laminated angleply rectangular plates. R. C. FORTIER and J. N. ROSSETOS 1974 International Journal of Solids and Structures 10. 14171429. Effects of inplane constrains and curvature on composite plate behaviour. S. SHARMA, N. G. R. IYENGAR and P. N. MIJRTHY 1980 International Journal of Mechanical Sciences 22, 607620. Buckling of antisymmetric cross and angleply laminated plates. S. SHARMA, N. G. R. IYENGAR and P. N. MURTHY 1980 American Society of Civil Engineers Journal of the Engineering Mechanics Division 106 (EMl), 161176. Buckling of antisymmetritally laminated crossply and angleply rectangular plates. Y. HIRANO 1979 American S0ciet.y of Mechanical Engineers Journal of Applied Mechanics 46, 233234. Buckling of angleply laminated circular cylindrical shells. K. P. SOLDATOS 1983 International Journal of Engineering Sciences 21, 217222. On buckling and vibration of antisymmetric angleply laminated circular cylindrical shells. K. P. SOLDATOS 1983 Quarterly Journal of Mechanics and Applied Mathematics 36, 20’722 I. Free vibrations of antisymmetric angleply laminated circular cylindrical panels.
132
K P. SOLDATOS
15. J. M. WHITNEY 1984 American Institute ofAeronautics and Astronautics Journal 22, 16411645. Buckling of anisotropic laminated cylindrical plates. 16. J. M. WHITNEY and N. J. PAGANO 1970 American Society of Mechanical Engineers Journal of Applied Mechanics 37, 10311036. Shear deformation in heterogeneous anisotropic plates. 17. C. W. BERT and T. L. C. CHEN 1978 International Journal ofSolids and Structures 14,465473. Effect of shear deformation on vibration of antisymmetric angleply laminated rectangular plates. 18. J. N. REDDY and N. D. PHAN 1985 Journal ofSound and Vibration 98, 157170. Stability and vibration of isotropic, orthotropic and laminated plates according to a higherorder shear deformation theory. 19. J. N. REDDY 1979 Journal ofSound and Vibration 66, 565576. Free vibration of antisymmetric, angleply laminated plates including transverse shear deformation by the finite element method. 20. R. D. MINDLIN 1951 American Society of Mechanical Engineers Journal of Applied Mechanics 18, 3138. Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates. 21. G. HERRMANN and I. MIRSKY 1956 American Society of Mechanical Engineers Journal of Applied Mechanics 23, 563568. Threedimensional and shelltheory analysis of axially symmetric motions of cylinders. 22. I. MIRSKY and G. HERRMANN 1957 Journal ofthe Acoustical Society ofAmerica 29, 1116 1129. Nonaxially symmetric motions of cylindrical shells. 23. G. B. WARBURTON and S. R. SONI 1977 Journal of Sound and Vibration 53, l23. Resonant response of cylindrical shells. 24. P. K. SINHA and A. K. RATH 1975 Aeronautical Quarterly 26,21 I218. Vibration and buckling of crossply laminated circular cylindrical panels. 25. S. B. DONG and F. K. W. Tso 1972 American Society of Mechanical Engineers Journal ofApplied Mechanics 39, 10911097. On a laminated orthotropic shell theory including transverse shear deformation. 26. Y. S. Hsu, J. N. REEDY and C. W. BERT 1981 Journal of Thermal Stresses 4, 155177. Thermoelasticity of circular cylindrical shells laminated of bimodulus composite material. 27. C. W. BERT and M. KUMAR 1982 Journal ofSound and Vibration 81, 107121. Vibration of cylindrical shells of bimodulus composite materials. 28. K. P. SOLDATOS 1985 Zeitschrtft fir angewandte Mathematik und Physik 36, 120133. On the theories used for the wave propagation in laminated composite thin elastic shells. 29. V. PANC 1975 Theories of Elastic Plates. Leyden: Noordhoff. 30. M. LEVINSON 1980 Mechanics Research Communications 7,343350. An accurate simple theory of the statics and dynamics of elastic plates. 31. C. W. BERT 1984 Composite Structures 2, 329347. A critical evaluation of new plate theories applied to laminated composites. 32. A. BHIMARADDI 1984 International Journal of Solids and Structures 20, 623630. A higher order theory for free vibration analysis of circular cylindrical shells. Journal 23,18341837. 33. A. BHIMARADDI 1985 American InstituteofAeronauticsandAstronautics Dynamic response of orthotropic, homogeneous, and laminated cylindrical shells. 34. J. N. REDDY and C. F. LIU 1985 International Journal of Engineering Science 23, 319330. A higherorder shear deformation theory of laminated composite shells. 35. K. P. SOLDATOS 1986 International Journal of Solids and Structures 22, 625641. On thickness shear deformation theories for the dynamic analysis of noncircular cylindrical shells. 36. K. P. SOLDATOS 1986 American Society of Mechanical Engineers Publication P119ND3, 2334, D. Hui, J. C. Duke, and H. Chung, editors, Chicago. Stability and vibration of thickness shear deformable crossply laminated noncircular cylindrical shells. 37. B. 0. ALMROTH 1966 American Institute of Aeronautics and Astronautics Journal 4, 134140. Influence of edge conditions on the stability of axially compressed cylindrical shells. 38. A. E. H. LOVE 1927 A Treatise on the Mathematical Theory of Elasticity. Cambridge University Press, fourth edition. 39. H. KRAUS 1967 Thin Elastic Shells. New York: Wiley. A comparison of some shell 40. K. P. SOLDATOS 1984 Journal ofSound and Vibration 97,305319. theories used for the dynamic analysis of crossply laminated circular cylindrical panels. 41. I. MIRSKY 1964 Journal oftheAcoustica1 Society ofAmerica 36,4151. Vibrations of orthotropic, thick, cylindrical shells. 42. K. SHIRAKAWA 1983 Journal of Sound and Vibration 91,425437. Effects of shear deformation and rotatory inertia on vibration and buckling of cylindrical shells.
ANGLEPLY
LAMINATED
133
C‘YLINDRIC‘AL PANELS
43. P. C. YANG, C. H. NORRIS and Y. STAVSKY 1966 Internationai Journal qf Solids and Structures 2, 66.5683. Elastic wave propagation in heterogeneous plates. 44. J.M. WHITNEY and C. T. SUN 1973 Journal ofSound and Vibration 30,8597. A higher order theory for extensional motion of laminated composites. 45. J. M. WHITNEY 1973 American Society of Mechanical Engineers Journal of Applied Mechanrcs laminates under static loads. 40, 302304. Shear correction factors for orthotropic 46. J. M. WHITNEY and C. T. SUN 1974 American Society af Mechanical Engineers Journal of Applied Mechanics 41, 471476. A refined theory for laminated anisotropic cylindrical shells. 47. C. T. SUN and J. M. WHITNEY 1974 Journal of the Acoustical Society of America 55, 1233 1246. Axisymmetric vibrations of laminated composite cylindrical shells. 48. L. V. KANTOROVICH and V. I. KRYLOV 1964 Approximate Methods in Higher Analysts. Groningen: Noordhoff. 49. S. CHENG and B. P. C. HO 1963 American Institute ofAeronautics and Astronautics Journal I, 892898. Stability of heterogeneous aeolotropic cylindrical shells under combined loading. SO. C. W. BERT, J.L. BAKER and D. M. EGLE 1969 Journal af Composite Materials 3, 480.499. Free vibrations of multilayer anisotropic cylindrical shells.
APPENDIX
A
For the shell theories based on the assumption of uniform 2.1), the components of the operational matrix [_Y] appearing follows: y, I = A 7 ),,Tl + &h(
)& L%( ).I,,
Y,,=Y9,,
Y,, = Y3, = A5GC,(
thickness shear (section in equation (12) are as
=A(;i,,S&,)(
),.,+
),,,,
(Al)
For the shell theories based on the assumption of a parabolic shear (section 2.2), the components of the operational matrix [p] (24) are as follows:
variation appearing
of thickness in equation
APPENDIX
B
For the shell theories based on the assumption of a uniform 2.1), the components of the submatrices appearing in equation (T,,),,
=
@id,, = (Tz,)r,= hmn~2(A,2+A,,)6,,6,,,
[(hm~)‘+(n~)‘A,,16,,6,,,
(T,3)r,s= 44Am7r&,@(n,
thickness shear (section (30) are as follows:
i)@(j, m),
(T,,),x = (Tdl)r\ =2Amn~‘B,,6,Jrn,,
(T,s)r\ = (T<,),, = [(Am~)2B,,+(n~)2B2,16,,6,, (T22)r.T= [6LT+Zk4&4+ (T&r\
(n7r)2A221L&,,
= 4~(n~)(A?z+~~7k4A44)0(i,
(T24)rr
=
(Ta2jrr
=
(T,,),, (T32),,
n)@(m,j), (n~)2B261~ni~mjr
[(AmrJ2&+
(T25)rr = 2Amn~2~26&r&,,
(Tulrs
(Amv)‘&,+
48LT$A‘k4A440(i,
=44AmrA,
[email protected](i,
n)@(j,
=4~n~(A22+~~7k4A44)0(i,
=
[d2Az2
+
(Am~)‘b&
(T3Jrs = (T43)rr = A2m~kS&SS,,&,,,
= [A2k&+
(Amrr)‘B,,
m), n)@(j,
+
m),
(n~)2LL16,i6,,
+4&7rB2,@(i,
(T35)rr= Anrrk,&48nrSm, +4+AmrB,,O(i, (T+d,
n)O(m,j),
n)@(_L m),
n)@(j,
+ (nr)2De61S,iS,i,
m),
.4NcLEPLY
(‘LJrr
LAMINATED
=
(TsJr~
=
CYLINDRICAL
[email protected],2
+
&hL,,
(T52)rs =2Amnx2B,,S,;G,j46,,~A~‘k,;i,,0(n,
i)O(j,
(T53)rs = AnrkdA448niSm, 4$~hm7r&O(n, (Tx)r,
=
0&2)r5
(Hdrr
=
=
(H52)re
= (Hx)r., = =
OUrs
[email protected](n,
U%drr
i)O(j,
m),
m),
=[A2k,A,,+(Amrr)2D,,+(n’rr)2Dz21Snrsm,,
(H,j)rr WxLs
135
PANELS
=
0+25),.5
Oh),,
=
=
0,
j)@(j, m), (&Jr\
(H,,),,
(TMrc
=
(H3s)r\
Of,,),,
=
=
0,
(H22)rc
=
OL)r~
= (Kn)rc
[email protected](i,
=,&&A,;,
n)@(m,j), (Bl)
= (H55)r\ = A2P2a,$rn,.
Here r=(jl)N+i,
(B2)
s=(ml)N+n,
and (B3)
‘sin(kr_y)cos(Izy)dy= I0
O(k,Z)=
where k and I are real integers. The column matrices A, B, C, D and E appearing in equation (30) contain the unknown coefficients a,,,,,, b,,, c,,, d,, and em,,, respectively, in such a way that AT={a,,,
42,.
. . , ulNy..
. , aM1,
*M2r..
BT=h,
612,.
. . , hrv,.
. , bM,,
ha..
CT={%,
cI2,.
DT={4,,
.
> GIN,
...,
CMlr
, UMN},
. > b,w},
cM2,....
CM,),
d,z,.
. . , d,,v,.
. . , dM,,
&a..
. , d,N},
e12,.
.
. . , eMI.
eM23..
. , eMN}.
ET={el,,
, elN,.
(B4)
For the shell theories based on the assumption of a parabolically distributed thickness shear (section 2.2), the components of the submatrices appearing in equation (30) are as follows: (Tii), (T,,),
=
[(hm~)2+(n~)‘A,,16,,6,,,
=46,r~[A(m~)2B,,+A~‘(n~)2B2,]O(n,
i)O(j,
+Amn~'(A,,+;i,,)6,,6,+4~LT~A~'n~B,,~(n
(T,3)r5 = 4dAmrA,,@(n, (TIJr,
i)O(j,
m) ii)@(j,m),
m)n~[3A(m~)2B,,+A~‘(n~)2B,,]6,,6,,,
= (Tdl)rs = 2Amnn’(&
%,)&J~,,
(TiS)r.T=(T~i)rY =[(Amrr)*(B,,B’,,)+(n~)‘(B2,BSh)IG,is,,, (T,,),,
=4S,,~[A(m~)‘B,,+A‘(n~)B,,]O(i,
n)O(m,
j)
+Amn~2(A,2+A66)~ni~m,+4~Lr~m~~lh~(m+j)O(i, (T22)r.s
=[(m~J2(A2&,+
SLT~“D~G)+
+168,,4mn7r2B2,@(i,
(nr)‘(A22+
~LT~‘A
n),
‘D22)ISniSrnj
n)0(m,j)+86,,~A~~‘n~B2,~(m+j)O(i,
n),
(T2dr, = (m~){B2,[S,,~2+3(n~)21+(Am~)2B,6}6,i6,, 4~n~{~22+6,,[(m~)2(~,2+2~66)+A~Z(n~)~~22]}O(i, 861.T~A~‘mn~TTIDh6CP(m+j)O(i,
n),
n)@(m,j)
136
K. P. SOLDATOS
CT,,),, =[(Amdz(&
~~,)+(n~)z(B2hBih)lSn,Sm,
+46,,~mn*‘(D,2Dl,+ij,,D~,)O(i,n)O(m,j) +46,,~(n~)(D,,D~,)~(m+j)8(i,
(Tz,),, = 2Amnv’( +A
&
f?&,)6,,6,,
n), +46,,4[h(mr)‘(~,,
‘(n7r)‘(D,,&)]@(i,
n)@(m,,j)
+4S,.~(m~)(D,,D~,)~(m+j)O(i, (T7,)rr =4dAmrA,
[email protected](i,
n)O(j,
(T32)rc= mr{&,[S1
n),
m)nr[3A(mr)‘fi,,+A
rd’+3(W’l
‘(n~)~&,]&,8,,,,,
+ (Am~)~&lf,d,,~,
+4~n57{A71+(SL,[(m57)2(L5,2+2Dhh)+A
‘nr{D,
[email protected](m+j)@(i,
461&h
&,I
‘(n~l’L])@(i,
n)
[email protected](n+i)@(,j,
n)O(j,
ml
m)},
(T,,),, =[~1A~2+A~(m*)‘D,,+2(mn~‘!‘(L),,+2D,,i +
A I(r~~)%‘~~]6,,,&~, 16dmnr’&O(
i, n)@( j, m 1,
(T34)r,=(m~)[(Am~)‘(D,,ij;,)+(n~)’(n,2Di,+20,,2D~,)1S,,6,,,, +4f$nx(B,,Bi,)O(i,
n)@(j, m), @,+2o,,2DA,)+A
(Ts5),., = (nr)[A(mr)‘(o,?+ 4c$Am77( & (T4z)r, =
 A&,)Wi,
I’( Blh
[(Amr
n)O(j,
+461TqU
(T43)r.,= (mn)[(Ama)‘(
m),
O&+D,,,D&JO(i,
n)@(m,ji
~“nrr(D,,iilr)~(m+j)O(i,
n),
o,, n;,)+(n7~)~(11j,U;~+2D~~2D;~)]6,,,6,,,,
4c#m77(&
(TJ_
B~,)8(i,n)H(m,j),
=[h’A,,+(hma)‘(D,,?DI,+
u;‘,)+(n~)‘(D,,20~,+D~,)]6,,,6,,,,
(Td5jr, = (T5Jr, = Amnk(fi,,20/,+ (T<,),, = 2Amnd +A
&
~;lz+o,,20~,fr56h)s,,,s,,,
8’J6,,,6,,,,
+46, +,b[hmr)‘(
‘(n~)~(D_~Di,)]g(n,
m),
(T53),,=n~[A(mrr)‘(D,,ij;2+2D,,2D~,)+A 4~$Am7~il?,,
‘(n~)‘(D~2Di~)]~nl~,,,,
I?~,)@(n, i)O(,j,
m),
2L)k,+ E&)t (nz):( &2Di2+
CT,,),, =[A2&,+(AmdU& = PO&~&,,,,
(HI1),, =4AmqT,O(n,
(H,,),, =4A(p, ijpdO(n, (Hz),,
= (P,,+26,+#+,
(H,,),,
= (Ed,\
(H4,Jr, =4(p,
n)O(j,
= A’m~(p~:P4)(S,l,fl,,,r %)@(i,
n)@(m,j),
D&)]S,,,S ,,,,,
i)@(j,m),
i)@(,j, ml,
(H2~)r,=4n~(p,+~,p2)0(i,
+ &dPX,,fL~r
(Hz5),, =4A[p,+~~,~(p,+cbPj)1(,(In)O(m,j),
(Hx2Jr5=4nr(p,+&)Wi,
h,,,  I?,,)
i)@(,j, m)
+46L,~A~~nsr(L52~151i?2)~(n+i)0(j,
(HI,),,
fXJ)]8,&,,
(mrf( &,  %JlbL,
 B’,,) +
+46,,~mn7r’(D,,
‘(nr)‘(l%
m),
(H,,),, =4hmrp,O(i,
n)O(m,j). n)@(j,
m t,
(H71),, =(P,,+~I[(Amrr)‘+(n~)~]}6,,6,,,,, CH7,),, = (H,?),, = A~n~(~~~p,)6,,6,,,,, (IL),,
= (IL),,
=AZ(p2~~+‘~Ph)~,,,~nl,r
(H5?),, =4A[~,+~p~~(p~+~p4)]0(n,i)(*)(.j,m).
iUS)
ANGLEPLY
LAMINATED
Here, the subindices r and s and the factor (B3), respectively. Moreover, O(k+l) Finally, the column matrices expressions (B4), while
CYLINDRICAL
O(k, 1) are given by expressions
= I (l)k+‘,
k,l=
A,B and C appearing
DT=P,,,
Q?,...,
PANELS
137 (B?) and
I,2 ,....
(B6)
in equation
(30) are still as given in
D,N, . . . . D,,,,...,D,,\l,
ET=W,,, E,z,. . . , E,,,
. . , EL,,, E,,,, . . . , Ecvivl.
(B71